An inner automorphism is only an inner automorphism, but an inner endomorphism can be something strange

AN INNER A UTOMORPHISM IS ONL Y AN INNER A UTOMORPHISM, BUT AN INNER ENDOMOR PHISM CAN BE SOME THING STRANGE GEORG E M. BER GMAN Abstract. The inner automorphisms of a group G can be characte r ized within the category of groups without refer ence to group elements: they are precisely those automorphisms of G tha t can b e extended, in a functorial m anner, to all groups H given with homomorphisms G → H. (Precise statemen t in § 1.) The group of such extend ed systems of automorphisms, unlike the group of inner automorphisms of G itself, is alwa ys isomor phic to G. A similar characte r ization holds for i nner automorphisms of an asso ciative algebra R o ve r a ﬁeld K ; here the group of functorial systems of automorphisms is isomorphic to the gr oup of units of R modulo the units of K. If one lo oks at the ab ov e f unctorial extendibility prop erty for endomorphisms, rather than j ust auto- morphisms, then in the group case, the only additional example is the trivi al endomorphism; but in the K -algebra case, a construction unfamiliar to ring theorists, but kno wn to functional analysts, also arises. Systems of endomorphisms with the s ame functoriality property are examined in some other categ or ies; other uses of the phr ase “inner endomorphism” in the literature, some ov erlapping the one introduced here, are noted ; the concept of an inner derivation of an associative or Lie a l gebra is lo ok ed at from the same poin t of view, and the dual concept of a “co-inner” endomorphism is brieﬂy examined. Several op en questions are noted. Ove r view. Y ou can r ead this overview if you’d like to know the topics of the v arious s ections to come; but feel free to skip it if you’d prefer to plunge in, and let the sto ry tell itself. In § 1 , we motiv ate the approach of this pap er using the ca se of groups. W e o btain the c ha racteriza tio n of inner automorphisms of groups that is stated in the abstr a ct, a nd, mo deled on this, we deﬁne concepts of inner a uto morphism a nd inner endomo r phism for an ob j e ct of a genera l categor y . In § 2, these deﬁnitio ns are a pplied to asso cia tiv e unital a lgebras ov er a commut a tive ring K, and full characterizations of the inner a utomorphisms and endomorphisms are obtained in the case where K is a ﬁeld. In § 3, co un ter examples are given to the obvious g eneralizations of these results to ba s e ring s that are not ﬁelds, and the question of wha t the genera l inner automo rphism and endomorphism might lo o k like in that case is exa mined. In § 4 we pa us e to survey concepts that have app ear ed in the liter ature under the name “inner endomo r- phism”, with v arying degr ees of ov er lap with that of this note. § 5 contains some ea sy obser v ations on inner automorphisms and endomorphisms (in the s ense her e deﬁned) on a few other sorts of algebra ic ob j ec ts . In §§ 6- 8 we study inner derivations of asso c iative and Lie alge bras, and also inner endomorphisms of L ie algebras , pausing in § 7 to consider what the genera l deﬁnition of “inner deriv ation” sho uld b e. It is noted in § 9 that our concept o f inner endomorphism dualiz e s to one of c o-inner endomorphism , and we determine these for the ca tegory of G -sets, for G a gr oup. In § 10 we brieﬂy lo o k a t the ideas of this pap er from the p ersp ective of the theo ry of repre s ent a ble functors. 2010 Mathematics Subje ct Classiﬁc ation. Primary: 16W20. Secondary: 08B25, 16W25, 17B40, 18A25, 18C05, 20A99, 46L05. Key wor ds and phr ases. Group, associative algebra, Lie algebra; inner auto mor phism, inner endomorphism, inner d er iv ation; comma category . ArXiv URL: http://arXiv .org/abs/1001.1391 . I disco vered the main results of Theorems 1 and 6 sev er al decades ago, at which time I w as partly supp orted by a National Science F oundation gran t, but I cannot no w reconstruct the date, and hence the gran t n umber. I sp ok e on those results at the Jan uary , 2009 AMS-MAA Joint Meeting in W ashingt on D.C. . After publication of this note, up dates, er rata, r elated refer ences etc., if f ound, will be recorded at http://math.b erkeley.edu/ ~ gbergman /papers/ . 1 2 GEORG E M. BERGMAN Although w e were not able to obtain in § 3 a full description of the inner endomorphisms o f a n a sso ciative K -algebra when K is not a ﬁeld, we prove in a ﬁnal appendix , § 11, using the partia l r esults of § 3, that such inner endo mo rphisms a re a lw ays one-to-o ne . Op en questions ar e noted in §§ 1, 3, 6 a nd 8. I am indebted to Bill Arveson for helpful references, and to the r e feree for many useful sugges tions. 1. In ner automorphisms and inner endomorphisms of groups. Recall that an automorphis m α of a group G is called inn er if there exists an s ∈ G such that α is given b y conjuga tion by s : (1) α ( t ) = s t s − 1 ( t ∈ G ) . Given this deﬁnition, it ma y seem perverse to ask whether the condition that α b e inner can be char- acterized without sp eak ing of group elemen ts. Note, howev er, that the deﬁnition implies the following prop erty , whic h ca n indeed b e so stated: F or every homomorphism f of G into a gr oup H, there exists an automorphism β f of H making a commuting square with α : (2) G ✲ H f G ✲ H f . ❄ α ❄ β f (Namely , we can take β f to b e conjuga tio n by f ( s ) . ) Whether this prop erty a lo ne is equiv alent to α be ing inner, I do not k now; but the ab ov e conclusion can be s trengthened. Let α be as in (1), and for every gr oup H and homomor phism f : G → H let β f be, a s ab ov e, the inner automor phism of H induce d by f ( s ) . Then this system o f auto morphisms is “coherent”, in tha t for every commuting triangle of gro up homomorphisms (3) G ✟ ✟ ✟ ✯ f 1 ❍ ❍ ❍ ❥ f 2 H 2 H 1 ❄ h one ha s (4) β f 2 h = h β f 1 . Let us show that this streng thened statement is equiv alent to α be ing inner ; and that the family of morphisms β f do es what α a lo ne in gener al do es no t: it uniquely determines s. Theorem 1. L et G b e a gr oup and α an aut omorphism of G. Supp ose we ar e given, for e ach gr oup H and homomorphism (5) f : G → H, an aut omorphism β f of H , with the pr op erties that (i) β id G = α, and (ii) f or every c ommuting triangle (3) one has (4) . Then ther e is a unique s ∈ G such t hat for al l H and f as in (5) , one has (6) β f ( t ) = f ( s ) t f ( s ) − 1 ( t ∈ H ) . In p articular, α is inner. Thus, an aut omorphi s m α of a gr oup G is inner if and only if ther e exists such a system of automorphisms β f . Pr o of. T o inv estigate the system of maps β f , let us lo ok at their b ehavior on a “generic” elemen t. Since the domains of these maps ar e groups with homomor phisms of G into them, a generic member of such a group will b e the element x of the group G h x i obta ined by adjoining to G one additional e le men t x and no additional rela tions. (This g roup is the copro duct G ` h x i of G with the free g roup h x i o n one g e ne r ator; in g roup-theor ists’ language and nota tion, the free pro duct G ∗ h x i . ) INNER ENDOMORPHISM S 3 So letting η b e the inclusion map G → G h x i , consider the element β η ( x ) ∈ G h x i . By the structure theorem for copro ducts of groups, this can b e written w ( x ) , where w is a reduced word in x and the elements of G. No te that for any map f of G into a gro up H , and any element t ∈ H , we c a n form a triangle (3) with H 1 = G h x i , f 1 = η , H 2 = H , f 2 = f , and h ( x ) = t. (There is a unique such h making (3 ) commute, by the universal prop erty of G h x i . ) By (4), the element β f h ( x ) = β f ( t ) is equal to h β η ( x ) = h w ( x ) = w f ( t ) , where w f denotes the result of s ubs tituting for the elements o f G in the word w their ima ges under f . Thus, β f acts by car rying every t ∈ H to w f ( t ) . Conv ersely , starting with any element w ( x ) ∈ G h x i , the formu la β f ( t ) = w f ( t ) clearly gives a set map β f : H → H for each f as in (5), in such a way that (ii) ab ov e ho lds. T o deter mine when these s e t maps resp ect the group op eration, we should co nsider the eﬀect of the map induced by w ( x ) on the pr o duct of a generic p air of elements. So we now take the gro up G h x 0 , x 1 i g o tten by a djoining to G tw o ele men ts and no relations, let η b e the inclusion of G there in, a nd consider the relation β η ( x 0 x 1 ) = β η ( x 0 ) β η ( x 1 ) , i.e., (7) w ( x 0 x 1 ) = w ( x 0 ) w ( x 1 ) . When we transform the pro duct on the right-hand side o f (7) into a re duced word in x 0 , x 1 and nonidentit y elements of G, the only reduction that can o ccur is the simpliﬁcation o f the pro duct of the factors from G at the rig h t end of w ( x 0 ) and at the left end of w ( x 1 ); in particular, all o ccurrences of x 0 contin ue to o c c ur to the left of all o ccurr ences of x 1 . On the o ther hand, in the left side of (7), each o ccurrence of x 0 or x 1 is adjacent to an o ccurrence of the o ther. It is ea s y to deduce that w ( x ) ca n contain at mo s t o ne o ccurrence of x, and that such an x, if it o ccurs, must have exp onent +1 . Mor eov er, if there were no o ccurrences o f x, then the functions β f would b e consta nt, hence could not give automorphisms o f nontrivial gro ups H ; so w ( x ) must hav e the form s 0 x s 1 . Substituting int o (7), we ﬁnd that s 1 s 0 = 1; hence letting s = s 0 , we hav e w ( x ) = s x s − 1 . Thu s , the ma ps β f hav e the form (6). Mor eov er, distinct elements s give distinct words w ( x ) , hence give distinct sys tems of automorphisms, since they act diﬀerently o n x ∈ G h x i ; so such a system of auto morphisms determines s uniquely . Combining the above des cription of β f with condition (i), w e see tha t our origina l automor phism α is inner. This gives the “if ” direction of the ﬁnal sentence of the theo r em; the rema rks at the beginning of this section g ive “only if ”.  (In the a bove theorem, we did not explicitly assume commutativit y of the dia grams (2). But in v iew of condition (i), tha t co mm utativity r equirement, for given h, is the case of condition (ii) wher e H 1 = G, f 1 = id G , H 2 = H , h = f 2 = f . ) F or ﬁxed G o ne can clearly compo se tw o coherent sy stems of automorphisms of the sort c onsidered in Theorem 1 to get a nother such system; a nd we see from the theor em tha t under comp ositio n, these systems form a gr oup isomo rphic to G. There is an elegant formulation of this fact in terms of comma catego ries. Recall that for any ob ject X of a category C , the categor y whose o b jects a re ob jects Y of C given with mor phis ms X → Y , and whose mo rphisms are c o mm uting triang les analog ous to (3), is denoted ( X ↓ C ) (called a “comma category” bec ause of the older notation ( X , C ); see [21, § I I.6]). A s ystem of maps β f as in Theo rem 1 a sso ciates to each ob ject f : G → H o f the comma categor y ( G ↓ Group ) an automor phism, not o f that o b ject, but of the g roup H ; i.e., of the v alue, at tha t ob ject f : G → H, of the for getful functor ( G ↓ Group ) → Group . Our condition (4) s ays that these automorphisms β f should tog e ther compr ise an automo r phism o f that forgetful functor . In summar y , Theorem 2. F o r any gr oup G, the automorphism gr oup of the for getful functor U : ( G ↓ Group ) → Group is isomorphic to G, via the map taking e ach s ∈ G to t he automorphism of U given by (6) .  In the pro of o f Theorem 1 we use d the assumption that α and the β f were automor phisms, rather than simply endomor phisms, only o nce; to exclude the ca se where w ( x ) c ontained no o ccurr e nces of x. In tha t case, (7 ) for ces w to be the identit y element, whence the β f are the tr iv ial endomorphisms, ε ( t ) = 1 . So we ha ve Corollary 3 (to pro of of Theorem 1 ) . If in the hyp otheses of The or em 1 one everywher e substitutes “en- domorphi sm ” for “ automorphism”, then the p ossibilities for ( β f ) ar e as state d ther e, to gether with one 4 GEORG E M. BERGMAN additiona l c ase: wher e every β f is the trivial endomorphism of H. In the language of The or em 2, the en- domorphi sm monoid of the for getful fu n ctor ( G ↓ Group ) → Group is isomorphic to G ∪ { ε } , wher e ε is a zer o element.  (There is nothing exo tic ab out the trivial endomorphism; so the s econd half of the title of this note do es not apply to the catego ry of gr o ups.) Let us abstr act, and name, the concepts we hav e b een using. Deﬁnition 4. If X is an obje ct of a c ate gory C , then an endomorphism ( r esp e ctively automorphism ) of the for getful functor ( X ↓ C ) → C wil l b e c al le d an extended ( or if ther e is danger of ambiguity, “ C -extende d” ) inner endomorphi s m ( r esp., inn er aut omorphi s m ) of X . An ex t ende d inner endomorphism or automorphism wil l at times b e denote d ( β f ) , wher e f is understo o d to run over al l f : X → Y in ( X ↓ C ) , and the β f ar e t he c orr esp onding endomorphisms or automorphisms of the obje cts Y . An endomorph ism or automorphism of X wil l b e c al le d inner if it is the value at id X of an ext en de d inner endomorphism or automorphism of X . When ther e is danger of ambiguity, one may add “in t he c ate gory-the or etic sen se” and/or “with r esp e ct to C ”. So, like “mono morphism” and “ e pimo rphism”, the term “inner ” will acquire a certain tension b etw een a pre-existing s e ns e and a categor y-theoretic sense, which will ag ree in many , but not necessar ily in all cas es where the for mer is deﬁned. In subsequent sections we will study the categor y-theoretic concept in s everal other c a tegories. W e remar k that if C is a legitimate ca tegory (i.e., if its hom-sets C ( X , Y ) are sma ll se ts – in c la ssical language, se ts rather than prop er class e s – but if its ob ject-set may b e large), then the mono ids o f endo- morphisms, resp ectively , the g roups o f automorphis ms, of the for getful functor s ( X ↓ C ) → C ar e no t, in general, smal l monoids o r groups. How ever, there is a set-theoretic approach that handles suc h s ize-problems elegantly; see [5, § 6.4] (cf. [21, §§ I.6 -7]). Aside from this point, these constructions behave very nicely: if f : X 1 → X 2 is a morphism, it is ea sy to see that an extended inner e ndo morphism or automorphism o f X 1 induces via f an extended inner endomo rphism or automorphism of X 2 (in contrast with the behav- ior of ordinary endomor phis ms, automor phisms, and inner automorphis ms ); thus, these constructions give functors fro m C to the catego ries of (po ssibly lar ge) monoids and gro ups . (Howev er, we sha ll not us e this observ ation b elow.) W e e nd our co nsideration of these co ncepts in Group by r ecording a question mentioned ab ov e, gener- alized fro m automor phisms to endomor phis ms. Question 5. If α is an endomorphi sm of a gr oup G, such that for e ach obje ct f : G → H of ( G ↓ Group ) ther e exists an endomorphism β f of H making t he diagr am (2) c ommu te, must α t hen b e inner in the sense of Deﬁnition 4; i.e., is it then p ossible to cho ose su ch endomorphisms β f so as to satisfy (4) for al l c ommuting triangles (3) ? ( By the pr e c e ding r esult s , this is e quivalent to: Must α either b e an inn er automorphism in t he classic al sense, or the trivial endomorphi s m ? ) 2. The case of K -algebras. Let us now consider the same q uestions for ring s. Let R ing 1 denote the categ ory of all asso ciative unital rings. A general diﬃculty in the study of universal constructions in this category is the non tr iviality of the multilinear alg e bra of a belia n groups, i.e., Z - mo dules . Often things ar e no w or se if we generalize o ur considera tions to the c a tegory R i ng 1 K of asso ciative unital algebras over a general commutativ e ring K , and they then b ecome muc h better if we assume K a ﬁeld. Below, we shall b eg in the analy sis of inner endomorphisms of K -algebr as for K a genera l commutativ e ring; then, ab out ha lf-wa y through, we will hav e to restrict ourse lves to the case where K is a ﬁeld. In the next section we will exa mine what versions of our result mig h t b e true for ge ne r al K . So let K b e a ny commutativ e ring (where “ asso ciative unita l” is under sto o d), a nd R any nonzero ob ject of Ri ng 1 K . W e will ag ain use generic elemen ts. The extension of R by a single generic elemen t x in R ing 1 K has the K -mo dule decomp osition (8) R h x i = R ⊕ ( R xR ) ⊕ ( R xR xR ) ⊕ . . . ∼ = R ⊕ ( R ⊗ R ) ⊕ ( R ⊗ R ⊗ R ) ⊕ . . . . INNER ENDOMORPHISM S 5 Here the tenso r pr o ducts are as K -mo dules. T enso r pro ducts ov er K will b e almost the only tensor pro ducts used in this note, so we make the conv ention that ⊗ , without a subscript, denotes ⊗ K . The extensio n of R by t wo g eneric elements similar ly has form (9) R h x 0 , x 1 i = M n ≥ 0 i 1 , .. ., i n ∈{ 0 , 1 } R x i 1 R . . . R x i n R ∼ = M n ≥ 0 i 1 ,...,i n ∈{ 0 , 1 } R ⊗ R ⊗ . . . ⊗ R ⊗ R . Exactly as in the pro of o f Theo rem 1, every ex tended inner endomorphism of R will b e deter mined by the ima g e under it of x ∈ R h x i , which will b e so me element w ( x ) ∈ R h x i . And aga in, e very w ( x ) ∈ R h x i induces, for each ob ject f : R → S of ( R ↓ Ring 1 K ) , a set map of S into itself, sending each r ∈ S to w f ( r ) ∈ S, and these maps r esp ect mo rphisms amo ng such ob jects. So again, our task is to determine for which w ( x ) ∈ R h x i the induced se t- maps S → S are K -alg ebra homomorphisms. These maps will resp ect addition if and only if the r equired equation holds in the generic case, i.e., if and only if, in R h x 0 , x 1 i , (10) w ( x 0 + x 1 ) = w ( x 0 ) + w ( x 1 ) . I cla im that the only elemen ts w ( x ) ∈ R h x i satisfying (10) are those which a re homo geneous of deg ree 1 in x ; i.e., lie in the summand R xR of (8 ). I ndee d, if w ( x ) had a nonzer o comp onent in o ne of the hig her degree summands in (8 ), then on substituting x 0 + x 1 for x, one of the nonzero comp onents we w o uld get in the left-ha nd side of (10) would lie in a summand of (9) that inv o lved b oth x 0 and x 1 , while this is not true of the r ight-hand side of (10). On the other hand, if w ( x ) had a no nzero comp onent a in degre e zero, then the degree- z ero comp onent of the left-hand side of (10 ) would b e a, while that o f the r ig ht - hand side would be 2 a . So w ( x ) is homogeneous of degr ee 1; i.e., we may write (11) w ( x ) = n X 1 a i x b i for some a 1 , . . . , a n , b 1 , . . . , b n ∈ R. This necessar y condition for (10) to hold is suﬃcient as well; in fac t, it clearly implies that the functions induced by w ( x ) resp ect the K - mo dule structure. It remains to bring in the conditions that the op eratio n induced by w ( x ) resp ect 1 , and resp ect multi- plication. The former co nditio n says that (12) w (1) = 1 , i.e., (13) n X 1 a i b i = 1 , while the la tter co nditio n, (14) w ( x 0 x 1 ) = w ( x 0 ) w ( x 1 ) , translates to (15) n X i =1 a i x 0 x 1 b i = n X j =1 n X k =1 a j x 0 b j a k x 1 b k . T o study these conditions, let us now assume that K is a ﬁeld. In that case, if there is any K -linear depe ndence relation among the c o eﬃcient s a 1 , . . . , a n in (11), then w e c a n rewrite one of these elemen ts as a K -linea r combination of the r est, substitute into (11), collect terms with the same left-hand factor, and th us transform (11 ) in to an expres s ion of the sa me form, but with a smaller num b er of summands. W e can do the same if there is a K -linear rela tion among b 1 , . . . , b n . Hence, if we choose the expr ession (1 1) to minimize n, we g et (16) a 1 , . . . , a n are K -linearly independent, and b 1 , . . . , b n are K -linea rly indep endent. Now let A b e any K -vector-space basis of R containing a 1 , . . . , a n , and B any basis containing b 1 , . . . , b n . Then as a K -vector-space, the summand R x 0 R x 1 R ∼ = R ⊗ R ⊗ R of (9), in which the tw o sides of (15) lie, decomp oses a s a direct sum L a ∈ A, b ∈ B a x 0 R x 1 b. If for ea ch j a nd k we take take the 6 GEORG E M. BERGMAN comp onent of (15) in a j x 0 R x 1 b k ∼ = R, and dro p the outer factor s a j x 0 and x 1 b k , we get the equation in R, (17) δ j k = b j a k ( j, k = 1 , . . . , n ) . What this says is that if we write a for the row vector ov er R formed by a 1 , . . . , a n , and b for the column vector for med by b 1 , . . . , b n , then b a is the identit y matr ix I n . On the other ha nd, (13) says that a b is the 1 × 1 identit y matr ix I 1 . Thus, r egarding these vectors as describing homomo rphisms of right R -mo dules a : R n → R and b : R → R n , these rela tions say that a a nd b cons titute an isomor phism (18) R n ∼ = R as r ight R -mo dules . F or many so rts of rings R (e.g., any ring admitting a ho momorphism in to a ﬁeld), (18) can only hold for n = 1 . In such cas es, a and b b eco me m utually in verse elements, so (11) takes the form w ( x ) = a x a − 1 , and our inner endomorphism is an inner a utomorphism in the classica l sense. The element a s uc h that w ( x ) = a x a − 1 is easily seen to b e determined up to a scalar factor in K , so the g roup o f extended inner automorphisms o f R is isomor phic to the quotient gro up of the units o f R by the units of K . On the other hand, there are r ings R admitting isomorphisms (18 ) for n > 1 [20], [11], [1 2], [4]. If in such an R we take a row vector a and column vector b describing such an isomorphism, then by the ab ov e computations, the element w ( x ) = P a i x b i determines a n unfamilia r so r t of extended inner endomorphism of R . It is not har d to verify that this system o f maps can b e describ ed a s follows. Since (for a n y ring R ) the ring of endomorphisms of the right R -mo dule R n is isomorphic to the n × n matrix ring M n ( R ) , a mo dule isomor phism (1 8) yields a K -alg ebra isomo rphism M n ( R ) ∼ = M 1 ( R ) . Mor e- ov er, for every o b ject f : R → S of ( R ↓ R i ng 1 K ) , the vectors a and b over R induce vectors f ( a ) , f ( b ) ov er S s a tisfying the same relations, and hence likewise inducing isomorphis ms of matrix rings . The endo- morphism o f S induced by w ( x ) ca n now b e describ ed as the comp osite (19) S diag . − − − → M n ( S ) (( r ij )) 7→ P f ( a i ) r ij f ( b j ) − − − − − − − − − − − − − − − − − → ∼ = S . Since the right-hand arrow in (19) is bijective, the co mpo site arrow will, like the left-ha nd arr ow, alwa ys b e one-to-one , but will not b e surjective for any nonzer o S unless n = 1; so the la tter is the only ca se wher e the a bove co nstruction g ives aut omorphisms of the alg ebras S. These obs erv ations are summarized b elow, along with a ﬁnal asser tion which the r eader should not ﬁnd hard to verify , which corr e spo nds to a description of the degre e of no n uniq ue ne s s of the expr ession fo r an element w = P a i ⊗ b i in a tensor pro duct of K -vector-spaces , when written using the smallest num b er of summands (the rank of the element as a tensor); equiv alen tly , using K -linea r ly indep endent a i and b j . Note that (17), which we deduced using those conditions of K - linear indep endence, clear ly also implies them. Theorem 6. L et K b e a ﬁeld, and R a nonzer o K -algebr a. Then for every ext ende d inner automo rphism ( β f ) of R, ther e is an invertible element a ∈ R, unique up t o a sc alar factor, s uch that for e ach f : R → S, the aut omorphism β f of S is given by c onjugation by f ( a ) . Mor e gener al ly, e ach extende d inner endomorphism of R has the form (19) for a p air ( a, b ) , wher e for some n, a = ( a i ) is a length- n ro w ve ctor over R , and b = ( b i ) a height- n c olumn ve ctor, satisfying (13) and (17) , e quivalently, describing an isomorphism (18 ) . Two such p airs of ve ctors ( a, b ) and ( a ′ , b ′ ) , asso- ciate d with inte gers n and n ′ r esp e ctively, determine t he same extende d inner endomorphism if and only if n = n ′ and ther e exists some U ∈ GL( n, K ) such that (20) a ′ = a U, b ′ = U − 1 b.  The conclus io n n = n ′ in the a bove r esult follows from the uniqueness o f w ( x ) , and hence o f its r ank as a member of R ⊗ R ; but let us note a wa y to see it directly , and in fact to see that n is determined by the v alue o f o ur extended inner endo morphism at a n y no nzero ob ject f : R → S of ( R ↓ Ring 1 K ) . F rom (19) we see that the c entr alizer in S o f the image of our extended inner endo morphism will b e isomor phic to M n ( Z ( S )) as a Z ( S )-a lg ebra, where Z ( S ) is the center of S. In pa rticular, it will be free of rank n 2 as a mo dule over Z ( S ); and free mo dules ov er c o mm uta tive ring s hav e unique rank. W e hav e noted that (19) shows that ev er y extended inner endomorphism ( β f ) of R consists of one-t o-one endomorphisms β f . This too can be seen fr om elementary co ns iderations: Any K -algebr a S can b e embed- ded in a simple K -algebr a T ; and any endomo rphism of S aris ing from an extended inner endomo rphism INNER ENDOMORPHISM S 7 of R will then extend to an endomorphism of T , which nec e s sarily has trivial kernel. (The embeddability of any K -alge br a in a simple K -alg ebra was pr ov ed in [9, Corollar y 1 and Remar k 2]. A diﬀerent metho d of getting suc h an embedding, noted for Lie algebra s in [2 4, Theor em B], is also applicable to asso cia tiv e algebras .) It is not hard to add to Theorem 6 the neces sary and suﬃcien t condition for t wo extended inner endo- morphisms of R as in the ﬁnal statement to agree, not necessar ily g lobally , but at R, i.e., to determine the sa me inner endomorphism of R. The condition ha s the same form as (20), but with U now taken in GL( n, Z ( R )) . F or n = 1 , this is the exp ected condition that the conjugating e le men ts diﬀer b y an inv er tible central factor in R . (F or the reader fa miliar with [6, Chapter I I I] we remark that ( R ↓ Ri ng 1 K ) is the categor y there ca lled R - Ring 1 K , and that the R h x i o ccurring in the ab ov e a rguments is the under lying a lgebra o f the co algebra ob ject r epresenting the for getful functor R - Ring 1 K → Ring 1 K . Since the v alues of tha t for getful functor hav e, in particula r, additive gr oup structures, the functor can b e re garded as Ab -v alued, so by [6, Theorem 13 .15 and Corollary 14 .8], its r e pr esenting K -a lg ebra is freely generated over R by an ( R, R )-bimo dule. This is the R x R ∼ = R ⊗ R of (8). O ur extended inner endomor phisms of R corresp ond to endomor phisms of R h x i as a co-ring . Since these ar e in particular co-a belia n-group endo morphisms, they will be induced by bimo dule endomorphisms of R x R ; this is the con tent of (11). Our subsequent ar g ument s determine when such an endomorphism resp ects the c ounit a nd comultiplication of R h x i . ) 3. Wha t if K is not a field? F or a gener al commutativ e ring K and an arbitrar y o b ject R of Ring 1 K , a n y vectors a, b over R that s atisfy (13 ) and (17) will still yield an e lemen t w ( x ) = P n 1 a i x b i inducing an extended inner endomor - phism (19) o f R in Ring 1 K ; but w e ca n no longer say that every ex tended inner endomo rphism ha s this form. As a n easy counterexample, if K is a dir ect pr o duct K 1 × K 2 of tw o ﬁelds, then Ring 1 K ∼ = Ring 1 K 1 × Ri ng 1 K 2 , and one can s how that a n y extended inner endomor phism of an ob ject R 1 × R 2 of Ring 1 K ( R i ∈ Ring 1 K i ) is determined b y an extended inner endo morphism of R 1 and an ex tended inner endomorphism of R 2 . Now if R 1 and R 2 are b oth no nzero, and if they r esp ectively admit extended inner endomorphisms ( β 1 ,f ) a nd ( β 2 ,f ) , asso cia ted with distinct p ositive integers n 1 and n 2 , then these to g ether induce an extended inner endomorphism of R which do es not hav e the form (19 ) for any n. F or a diﬀerent sor t of ex ample, suppo se K is a commutativ e integral domain having a nonpr incipal in- vertible ideal J, a nd let F b e the ﬁeld of fractions o f K . (Recall that an ideal J o f K is called inv ertible if it ha s an inv er se in the m ultiplicative mo no id o f fr actional ide als of K, tha t is, no nze ro K -submo dules of F whos e elements admit a common denominator . The integral domains K all of whose nonzero ideals are inv ertible are the Dedekind domains [2, Theorem 9.8]. Thus, a ny Dedekind domain that is not a PID has a no npr incipal invertible ideal J. ) Supp ose we form the L a urent p olynomial ring in one indetermi- nate, F [ t, t − 1 ] , and within this, let R b e the subring K [ J t, J − 1 t − 1 ] . Then in R h x i , the K -submo dule J t x J − 1 t − 1 ∼ = J ⊗ J − 1 ∼ = K is fre e on one g enerator, which we sha ll ca ll w ( x ) , and which we might write (in)formally as t x t − 1 , tho ugh t itself is not an element of R . One ﬁnds that w ( x ) satisﬁes (10), (12) and (14 ), and s o induces an extended inner endomo rphism; but “ t x t − 1 ” do es not have the form s x s − 1 for any inv e rtible element s ∈ R , so this extended inner endomor phism is not a s describ ed in Theo rem 6. Inci- dent a lly , this extended inner endomorphism ha s an inv er se, induced by “ t − 1 x t ”, so it is even an ex tended inner a uto morphism (showing that the ﬁr s t half of our title is no t quite true). In taking an example of maximal simplicity , we hav e ended up with a commutativ e R , so that the automorphism of R itself induced by the ab ov e extended inner automor phism is trivia l, a nd can be describ ed as conjugation by 1 ∈ R. T o avoid this, let us freely a djoin to the F -alg e br a F [ t, t − 1 ] a nother noncommuting indeterminate, u, g etting the a lgebra F h t, t − 1 , u i , and within this take R = K h J t, J − 1 t − 1 , u i . Then the automorphism of R induced b y “ t x t − 1 ” is now nontrivial, and is still not inner in the c lassical sense; in particular, it takes u to t u t − 1 , though co njugation by no inv ertible element of R ca n do this. Our general result for K a ﬁeld, and the ab ov e ex a mples for other sorts of K , ca n be subsumed in a common co nstruction: Supp ose P is a K -mo dule, and R a K - a lgebra having an iso morphism (21) a : P ⊗ R ∼ = − → R 8 GEORG E M. BERGMAN as right R -mo dules . (In the case where K w as a ﬁeld, P was a n n - dimensional vector space; in our K 1 × K 2 example, it w a s the mo dule K n 1 1 × K n 2 2 ; in the K [ J t, J − 1 t − 1 ] and K h J t, J − 1 t − 1 , u i examples, it is J. In this la st ca se, one ha s an isomorphis m (21) J ⊗ R ∼ = R be c ause J ⊗ R ∼ = J R = t − 1 R ∼ = R, the middle equality holding b ecaus e R is closed in F [ t, t − 1 ] under multiplication b y J − 1 t − 1 and J t. ) Such a map (21) yields, for every algebra S with a homomorphism R → S, a K -alg e bra homomorphism (22) S ∼ = End S ( S S ) P ⊗− − − − → End S ( P ⊗ S S ) a ⊗ R − − − − − → ∼ = End S ( S S ) ∼ = S. The K -mo dule P in this construction need not b e unique. F or instance, if we ta ke an example ba sed on an isomorphism (18), but where our R is an algebr a over some epimorph K ′ of K (in the categ o ry-theor e tic sense; e.g., a factor-ring o r a lo caliza tion), then rega rding R as a K -a lgebra, we co uld cho ose the K -mo dule P of (21 ) to b e either K n or K ′ n . In a ll the cases lo o ked at so far, our K -mo dule P either was, or (in the ab ov e paragra ph) could be taken to be, pro jective o ver K . But there are examples where this is imp os sible: Co nsider a n y integral do main K which has an epimor ph of the form K 1 × K 2 for ﬁelds K 1 and K 2 (e.g., Z has suc h homomorphic imag es). Then if we construct, as in the ﬁrst paragr aph of this section, an algebra R ov er K 1 × K 2 and an extended inner endomorphis m of R based on a K ′ -mo dule P = K n 1 1 × K n 2 2 with n 1 6 = n 2 , this canno t a rise from an example based on a pro jective K -mo dule. This follows from the fact that for a ﬁnitely g enerated pro jective mo dule ov er an integral domain K , the rank is constant as a function o n the prime spec tr um of K [1 0, Ch.2, § 5 , n o . 2 , Theor` eme 1, (a ) ⇒ (c)], [19, p.53 , Exer cise 22]. Question 7. If K is a c ommutative ring and R a nonzer o obje ct of R ing 1 K , c an every extende d inner endomorphi s m of R b e obtaine d as in (22 ) fr om a mo dule isomorphism (21) ? The nonuniqueness of the P in the ab ov e cons tr uction makes me dubious. W e saw in the pr eceding section tha t for K a ﬁeld, a ll inner endomor phisms o f K -algebr a s were one-to-one. In a n app endix, § 11 , we show that the same is true for any K. 4. Other concepts of “inner endomorphism” in the litera ture. A MathSciNet sear ch for “inner endo morphism” leads to a num b er of concepts, some o f which hav e int er esting overlaps w ith the one we have b een studying. A str iking case, to whic h we alluded in the abstract, comes from the theor y o f C ∗ -algebra s. If H is a Hilber t spa ce, and B ( H ) the C ∗ -algebra of b ounded op era tors H → H , it is shown in [1, Pr op osition 2.1] that every endo morphism of the C ∗ -algebra B ( H ) has a form analogo us to what we found in T he o rem 6, namely (23) A 7→ X V i A V ∗ i , where the V i are a (p ossibly inﬁnite) family o f iso metric embeddings H → H having mutually or thogonal ranges which sum to H , a nd V ∗ i is the adjoint of V i . Here is a heuristic sk e tch for the a lgebraist of why this is plausible. Since complex Hilbert space s lo ok alike except for their dimension, it is na tural to genera lize the pro blem o f characterizing endomorphisms of B ( H ) to that of character iz ing homomo r phisms B ( H 1 ) → B ( H 2 ) for tw o Hilb ert s paces H 1 and H 2 . If H 1 and H 2 are ﬁnite-dimensional, of dimensions d 1 and d 2 , then B ( H 1 ) a nd B ( H 2 ) are matrix alge br as M d 1 ( C ) a nd M d 2 ( C ) . T empora rily igno ring the C ∗ structure, we know that a C -alg e bra homomorphism M d 1 ( C ) → M d 2 ( C ) exists if and only if d 2 = n d 1 for some integer n, and that in this case, it can b e gotten by writing C d 2 as a direct sum o f n copies o f C d 1 , and letting M d 1 ( C ) act in the natur al wa y on each of these. If a 1 , . . . , a n : C d 1 → C d 2 are the chosen embeddings and b 1 , . . . , b n : C d 2 → C d 1 the corr esp onding pro jections, the induced map M d 1 ( C ) → M d 2 ( C ) is given by (24) r 7→ X a i r b i . If one wan ts this to b e a ho momorphism of C ∗ -algebra s, one ha s the additional req uirement that the a i each map H 1 int o H 2 isometrically , with ortho gonal images; the pro jections b i will then b e the adjoints of the a i . Now if instead o f ﬁnite-dimens io nal Hilbert spaces we take an inﬁnite-dimensional Hilber t s pa ce H , and let H 1 = H 2 = H , then for b oth ﬁnite a nd inﬁnite n, there exist expressio ns of H a s a dir ect sum (in the inﬁnite ca se, a completed direct s um) of n copies o f itself. The r esult of [1] says that all e ndo morphisms of INNER ENDOMORPHISM S 9 B ( H ) are expres sible essent ia lly as in the ﬁnite dimensio nal case, in terms of such direct sum deco mpo sitions of H . F or R a ny C ∗ -algebra , not necess arily of the form B ( H ) , a family of elements V 1 , . . . , V n ∈ R ( n < ∞ ) satisfying the C ∗ -algebra relations corr esp onding to the conditions stated following (23) is equiv alent to a homomorphism into R of the C ∗ -algebra pres e nted by those gener ators and rela tions; this C ∗ -algebra is denoted O n . The ob jects O n are called C untz algebras , having b een introduced by J. Cun tz [13]. Since the ab ov e co nstruction with n = 1 gives inner a uto morphisms o f R in the classical sense, endomorphisms o f the form (23) in a gener al C ∗ -algebra (where they are not in general the only endo morphisms) are ca lle d inner endo morphisms. (F or n = ∞ , thing s are not a s neat. Though in B ( H ) , the inﬁnite sums (23) conv er g e in a to po logy obtained fro m the Hilb ert space H , this is not the top ology arising fr om the C ∗ -norm on B ( H ) . In deﬁning the C ∗ -algebra O ∞ one has to omit the rela tion P V i V ∗ i = 1 , bec ause the inﬁnite s um will not conv er ge; and maps o f this ob ject into a C ∗ -algebra R do not induce endo morphisms of R, though they are still of int er est.) The next co ncept I will describ e is not called an “inner endomorphism” by the author who studies it, though it did turn up in a Ma thSciNet sea rch for that phrase . (In the pap er in question, “inner endomor - phism” is used for “ endomorphism of a subalg ebra”.) Namely , in [23], if A is a n algebra in the sense of universal algebr a, a termal endomorphism of A means an endomo rphism α which is ex pr essible by a t erm in o ne v aria ble x, i.e., in the notation of this no te, a w o rd w ( x ) in the op erations of A and constants tak e n from A. Note that if such a word w ( x ) deﬁnes an endo morphism o f A, i.e., if the set map it determines resp ects all op era tions of A, then this fac t is equiv alent to a family o f identities in the op erations of A and the constants o ccurr ing in w . If V is so me v ar iet y c ontaining A, thos e identities need no t be satisﬁed by all mem b er s of ( A ↓ V ) , so w may no t deﬁne what we a re calling a V -extended inner endomorphism of A. How ever, if w e r egard ( A ↓ V ) as a v ariety , with the images of the elements of A as new zer oary op erations, then the identities named will deﬁne a subv ariet y V 0 ⊆ ( A ↓ V ) , on which w ( t ) do es induce an extended inner endo mo rphism of id A , and hence a n inner endomorphism of A. The phrase “ inner endo mo rphism” ha s in fact b een used in the theo ry of s emigroups [25], [27] to descr ib e some pa rticular clas ses o f what [23] calls ter mal endomorphisms. A diﬀerent use of the phrase “ inner endomorphis m” has o ccasiona lly been made in g roup theory . Obser ve that if G is a gro up, and α : G → G is a set map which in one or another se ns e ca n b e “approximated” arbitrar ily closely by endomor phisms, then in general, α will a gain b e a n endo morphism; but that if the approximating endomorphisms a re bijective, this do es not force α to b e bijective. In s uc h situations, if the approximating maps are inner automorphisms, α has bee n called an “inner e ndo morphism”, preceded by some qualifying a dverb. Sp eciﬁcally , if one can ﬁnd inner a uto morphisms o f G that agr e e with α on a directed family of subgroups having G as union (though the conjuga ting elements need not belong to the corres p onding subgr oups, so that α nee d not carr y those s ubgroups onto themselves), then α is ca lled (in [3], and [1 4, p. 20 1 , starting in parag raph be fo re Theorem 5 .5.9]) a “lo cally inner endomo rphism”, while if α induces inner a utomorphisms on a cla ss of homomor phic images of G tha t separates p oints, it is calle d in [7] a “residua lly inner endo morphism”. In the s a me spirit, [18] c a lls a top ological limit of inner a utomorphisms of a C ∗ -algebra an “ asymptotically inner endomorphism” (a usage a pparently unrelated to the sense of “inner endomorphism” o f a C ∗ -algebra descr ibe d ab ov e). On a somewhat related theme, [1 6] takes a ﬁnite-dimensiona l asso ciative unital alg ebra A over a ﬁeld K, with K -vector-space basis { u 1 , . . . , u n } , forms an extensio n ﬁeld K 0 of K by adjoining n algebr aically independent elements, use s these as co eﬃcients in forming a “g eneric” element of the K 0 -algebra A ⊗ K 0 , and notes that this element will necessa r ily b e inv ertible, so that conjugation b y it may b e thought of a s a “g eneric” inner automor phism o f A. It is then no ted that for certa in elements a ∈ A, the sp ecializa tion of o ur indeterminates to the co eﬃcie n ts of the u i in a may turn the ab ov e conjugation map in to a map that is everywhere deﬁned on A, even if a itself was not invertible. (In tuitively , the map obtained by that sp ecialization is approximated by the o per ations of conjugation b y nearby invertible elemen ts .) The resulting maps are endo mo rphisms, but exa mples ar e given showing that they may not b e automorphis ms , a nd they are named “inner endomo rphisms” of A. 10 GEORG E M. BERGMAN I don’t see a dire ct relatio n betw ee n the concepts cited in the last tw o parag raphs a nd those o f this pap er . How ever, p onder ing the idea of [16], in w hich one p erfor ms a conjugation r 7→ a r a − 1 for which, fro m the po in t of view of A, the pair ( a, a − 1 ) “ do esn’t quite exis t” , help ed lea d me to the exa mple of the preceding section, in which a conjugating element t w a s put out o f r each by multiplying by a no nprincipal inv ertible ideal J ⊆ K. I will note ano ther use of “inner” in the litera ture, not re stricted to endomor phis ms , at the end o f § 7 . W e now return to inner endomorphisms in the sens e of Deﬁnition 4. 5. E xtended inner endomorphisms in other ca tegories of al gebras – some easy obser v a tions. W e hav e examined e x tended inner endomorphisms in Group and Ring 1 K . What abo ut other catego ries of algebra s? In the category Ab o f ab elia n groups (which we will write a dditiv ely ), the result of a djoining a “generic” element x to an ob ject A is A ⊕ h x i , each element w ( x ) of which ha s the form a + nx for unique a ∈ A and n ∈ Z . Clearly , the system of op eratio ns induced by this element will resp ect the group op erations of a rbitrary o b jects of ( A ↓ Ab ) if and o nly if a = 0; so here the genera l extended inner e ndomorphism is given by m ultiplicatio n by a ﬁxed integer n ; it will be an extended inner automor phism if a nd only if n = ± 1 . These a re very diﬀerent fr o m the ex tended inner endomo rphisms of the same g roup A in the la rger category Group . Note that the ab ov e extended inner endomorphisms o f A do not rea lly dep end on A. Thoug h we are lo ok- ing at them as endomorphisms of the forg etful functor ( A ↓ Ab ) → Ab , they a re induced b y endomorphisms of the ident ity functor of Ab . W e might ca ll such op erations absolute endo morphisms. W e can answer in the negative the analog of Questio n 5 with Ab in pla c e o f Group . Let p b e a prime, and let A = Z p ∞ , the p -tor s ion subgroup o f Q / Z . Reca ll that this a be lia n gro up is injectiv e, that its no nzero homomorphic imag es ar e all iso morphic to it, and that its endomo rphism ring is canonically is omorphic to the ring of p -adic integers. It is easy to see that the action of each p -a dic integer c on A makes a co mm uting square with the action of c on every homo mo rphic imag e f ( A ) . Now if f is a homomorphism of A in to any ab elian group B , the injectivity o f f ( A ) implies tha t B c an b e de c ompo sed as f ( A ) ⊕ B 0 ; henc e the a ction of c on f ( A ) can b e extended to an action on B ; e.g ., by using the identit y o n B 0 . It follows that all the endomorphisms o f A (including its uncountably many auto morphisms) hav e the o ne- B - at-a- time extendibility pro p er ty analogous to the hypothesis of Ques tion 5, thoug h we ha ve seen that only those corres p onding to m ultiplica tio n by integers are inner, as deﬁned in Deﬁnition 4 . Hence in Ab , the one- B -a t-a-time ex tendibilit y pr op e rty is strictly weaker than the functorial extendibility pro per t y by which w e hav e deﬁned inner endomo rphisms and automo r phisms. It would b e interesting to inv estigate inner automor phisms and endomorphis ms in still o ther v arieties of groups. In the categ ory of c ommu ta tiv e rings, it is not hard to v er ify that Z has no nontrivial extended inner endomorphisms. On the o ther hand, Z /p Z has, fo r every p ositive integer n, the e x tended inner endo- morphism given by exp onentiation by p n (the n -th power of the F rob enius map). These endomorphisms are trivial o n Z /p Z itself; but o n every other integral do main of characteristic p, the F rob enius map is a nontrivial inner endomorphism. If A is a n ob ject of the v ariety of abelia n se migroups (wr itten m ultiplicatively), and e an idempo tent element of A, then multiplication by e is an inner endomor phism; the same is tr ue in the ca tegory of nonunital commutative rings. Similar ly , if D is an ob ject of the category of distributiv e lattices, then for any a , b ∈ D , the op erator s a ∨ − , b ∧ − , and a ∨ ( b ∧ − ) a re inner endomor phisms. If A is an ob ject of the categ ory of all semigroups (not necessar ily ab elian), and e is a c entr al idemp otent of A, then the word w ( x ) = e x gives a termal endomorphism of A in the sense of [23] (see preceding section), but not an inner e ndomorphism in our sense. Howev er, following the the idea noted in that s e ction, if w e form the subv ar iety of ( A ↓ Semigroup ) deﬁned by the identit y ma king the image of e central, then w ( x ) = ex do es determine an inner ex tended endo morphism in that categ o ry . The analo gous obser v ations hold for nonunital comm uta tiv e rings , and for not necess arily distributive la ttices. INNER ENDOMORPHISM S 11 6. Deriv a tions of associa tive algebras. Alongside inner automor phis ms of groups and rings, there is another pair of cases where the mo diﬁer “inner” is classical: inner deriva t ions of a sso ciative and Lie algebra s. W e shall examine the case of ass o ciative algebras in this sectio n, that of Lie alg e bras in § 8. If K is a commu ta tiv e ring and R an ob ject o f Ring 1 K , we r e c all that a derivation of R as a K -alg ebra means a set-ma p d : R → R satisfying (25) d ( r + s ) = dr + ds ( r, s ∈ R ) , (26) d ( c r ) = c dr ( c ∈ K, r ∈ R ) , (27) d ( r s ) = d ( r ) s + r d ( s ) ( r , s ∈ R ) . In particula r, for every t ∈ R, the map d deﬁned by (28) d r = t r − r t is a der iv ation of R, called the inner derivation induced by t, and written tr − rt = [ t, r ] . Such a n inner der iv ation d clearly has the ana log of the pr op erty of inner automorphisms of gro ups which we abs tracted in Deﬁnition 4; namely , that to every f : R → S in ( R ↓ R ing 1 K ) we c a n asso ciate a deriv ation d f of S, in such a wa y that (29) d id R = d, and that given tw o ob jects f i : R → S i ( i = 1 , 2 ) of ( R ↓ Ri ng 1 K ) a nd a mo rphism h : S 1 → S 2 in tha t category , w e hav e (30) d f 2 h = h d f 1 . What ab out the conv ers e? Giv en a sy s tem of deriv ations d f satisfying (30), let us , as in our investigation of automo rphisms a nd endomor phisms, lo ok a t their actio n on a g eneric ele men t. Let η : R → R h x i b e the natural inclusion a nd wr ite d η ( x ) = w ( x ) ∈ R h x i . As b efore , (2 5 ) implies that (31) w ( x ) = n X 1 a i x b i for some a 1 , . . . , a n , b 1 , . . . , b n ∈ R, and conv ersely , this condition implies bo th (25) and (26). T o handle (2 7), we need, as b e fo re, an additional assumption; but this time we can get aw ay with muc h less than K b eing a ﬁeld. Let us merely assume that the ca no nical map K → R makes K a K - mo dule direct summand in R ; i.e., that ther e exists a K -mo dule- theoretic left inv erse ϕ : R → K to that map. Given such a ϕ, it is not hard to see that we can obtain fro m (31) an equatio n o f the same for m (pos sibly with n increas ed by 1) in which a 1 = 1 , while a 2 , . . . , a n ∈ Ker( ϕ ) . So let us assume that (31) has tho se prop erties. Let us now take the generic ins tance of (27 ), na mely , in R h x 0 , x 1 i , the eq uation (32) n X 1 a i x 0 x 1 b i = ( n X 1 a i x 0 b i ) x 1 + x 0 ( n X 1 a i x 1 b i ) . The tw o sides of this equation lie in R x 0 R x 1 R ∼ = R ⊗ R ⊗ R. Le t us apply ϕ to the leftmost of the three tensor factors , g etting an eq uation in x 0 R x 1 R, a nd take the right coeﬃcie nt of x 0 therein. This is an equation in R ⊗ R ∼ = R x 1 R, namely (33) x 1 b 1 = b 1 x 1 + n X 1 a i x 1 b i . Solving for the summa tio n, which is w ( x 1 ) , a nd writing x in place o f x 1 , we get (34) w ( x ) = x b 1 − b 1 x . Hence, each map d f is the inner deriv atio n, in the clas sical se nse, determined by f ( b 1 ) . W e summar ize this res ult b elow. The “ unique up to . . . ” assertion in the co nclusion is obtained by noting that an element b ∈ R satisﬁes b ⊗ 1 − 1 ⊗ b = 0 in R ⊗ R = ( K ⊕ K er( ϕ )) ⊗ ( K ⊕ K er( ϕ )) if and only if the comp onent of b in Ker( ϕ ) is 0; i.e., if a nd only if b ∈ K. 12 GEORG E M. BERGMAN Theorem 8. L et K b e a c ommut ative ring and R a K -algebr a, and supp ose we have a function asso ciating to every f : R → S in ( R ↓ R i ng 1 K ) a derivation d f of the K -algebr a S, su ch that (30) holds for every morphism h of ( R ↓ R ing 1 K ) . Supp ose also that the c anonic al map K → R has a K -mo dule-the or etic left inverse. Then ther e exists b ∈ R , unique u p to an additive c onstant fr om K , such that for e ach f : R → S, d f is t he inner derivation of S induc e d by f ( b ) .  W e can push this a bit further. Instead of assuming that the canonica l map K → R has a K -mo dule left inv erse, assume the K - algebra structure o n R extends to a K ′ -algebra str uctur e for some epimor ph K ′ of K in the category of commu ta tive rings, and that the map of K ′ int o R has a K ′ -mo dule left inv erse. (This is the same a s a K - mo dule left inv er se to the latter map. On the other hand, when the e pimorphism K → K ′ is not a n iso mo rphism, the ma p of K its e lf into R cannot hav e a K -mo dule left inverse.) The n we can apply the ab ov e theore m to R as a K ′ -algebra ; moreov e r , it is not hard to show that ( R ↓ R ing 1 K ) ∼ = ( R ↓ Ri ng 1 K ′ ); so the characteriza tio n by Theore m 8 o f such systems of deriv atio ns pa rametrized by ( R ↓ Ring 1 K ′ ) gives the same r e s ult for sys tems o f deriv ations pa r ametrized by ( R ↓ Ring 1 K ) . Note, how ever, that the element inducing the system will b e unique up to an additive constant in K ′ , rather than in K . I know of no example showing the need for any version of the mo dule- theoretic hypo thesis of Theo rem 8 for the exis tence half of the conclus io n. So we as k Question 9 . If K is a c ommut ative ring and R an asso ciative unital K -algebr a, mu st every system of derivations ( d f ) satisfying (30) b e induc e d, as ab ove, by an element b ∈ R ? 7. H ow should one define “extended inner deriv a tion”? W e would hav e stated Theor em 8 and Question 9 in terms of “extended inner der iv ations of R ” , if it were clear how to deﬁne this concept. W e could, of course, make a n a d ho c de ﬁnitio n o f the phrase, a s a sy stem of deriv ations d f satisfying the hypo thesis of those statements; but it would b e b etter if we could make it an instance of a gener al use of “extended inner — ”. Deriv ations ar e not, in an obvious way , mo rphisms in a ca tegory , so we cannot use Deﬁnition 4. B elow, we will no te several wa ys that deriv ations ca n be put in a more general co nt ex t, then take the one that se ems b est as the basis of our deﬁnition. Recall ﬁrst that there is a w ell-k nown characterization of deriv ations in terms o f algebra homomo r phisms. If R is a K -alg ebra, let I ( R ) denote the K -algebr a obtained by a djoining to R a central, s quare-zer o element ε (intuitiv ely , an inﬁnitesimal. F or mally , I can be describ ed as the functor of tenso ring ov er K with K [ ε | ε 2 = 0 ] . ) Then it is eas y to verify that a set-map d : R → R is a der iv ation if and o nly if the map R → I ( R ) given by (35) r 7→ r + ε d ( r ) is a K - algebra homomorphis m. Under this corres po ndence, the inner deriv ations, in the cla ssical sens e, corres p ond to the homomor phis ms obtained by co mpo s ing the inclus ion R → I ( R ) with conjuga tion by a unit o f the for m 1 + ε b ( b ∈ R ) . Using this characterizatio n of deriv ations, we could put our co ndition o n families of deriv ations d f int o category -theoretic languag e. But I do n’t s ee the resulting fra mework as ﬁtting a natura l wider class o f constructions. A second appro ach b egins by asking, “Since the common v alue of the tw o sides o f (3 0) is neither a deriv ation of S 1 , no r a deriv ation of S 2 , no r a ring ho mo morphism, what is it?” It is, in fact, what is known a s “a deriv ation from S 1 to S 2 with r e spe c t to the homomorphis m h : S 1 → S 2 ”; i.e., a set- ma p d : S 1 → S 2 which satisﬁes (25), (2 6), and the gene r alization of (27), (36) d ( r s ) = d ( r ) h ( s ) + h ( r ) d ( s ) . If, now, for every pa ir of K -algebr as S 1 , S 2 , we le t D ( S 1 , S 2 ) deno te the set of all pairs ( h, d ) , where h : S 1 → S 2 is a K -algebr a ho mo morphism and d : S 1 → S 2 a der iv ation with resp ect to h in the ab ov e sense, then D ( − , − ) b ecomes a bifunctor ( Ri ng 1 K ) op × R ing 1 K → Set , having a forgetful mo rphism ( h, d ) 7→ h to the biv ariant hom functor Hom : ( Ring 1 K ) op × Ri ng 1 K → Set . The set of der iv ations of a single K -alg ebra S is the inv er se image of the identit y endomo rphism of S under this forgetful ma p. INNER ENDOMORPHISM S 13 Again, howev er, I don’t know of a natural class o f construc tio ns wider tha n the deriv ations to which these observ ations g eneralize. Moreover, the conce pt of a n h -der iv atio n d : S 1 → S 2 for h a ring homo morphism is in turn a sp ecial cas e o f that of an ( h, h ′ )-deriv ation, for h, h ′ : S 1 → S 2 t wo homomorphisms ; such a deriv ation is a map satisfying (25), (26), and (37) d ( r s ) = d ( r ) h ′ ( s ) + h ( r ) d ( s ) . In this context, we again ha ve the concept of the inner der iv ation induced b y an element b ∈ S 2 , namely the o pe r ation (38) d r = b h ′ ( r ) − h ( r ) b. How this ge ne r alization might relate to our concept of “ extended inner der iv ation” is not clear . A third approach is to start w ith any v ar iety V of algebr as in the sense of universal algebr a (i.e., the class o f all sets given with a family of o per ations o f sp eciﬁed arities, satisfying a speciﬁed set of identities [5, Chapter 8]), and supp ose tha t we a re interested in endomaps m o f the underlying sets o f ob jects of V which satisfy a c e r tain set of identities in the op er a tions of V and the inputs and outputs of m. (E.g., if the v ar ie t y is Ring 1 K and the ma ps are the deriv ations, the identities are (25 ), (26 ) and (2 7 ).) F o r ev e r y A ∈ V , le t M ( A ) denote the set of all such ma ps, and let us call these the M -maps of A. Then we may deﬁne an ext ende d inn er M -map of A as a w ay of asso ciating to e a ch f : A → B in ( A ↓ V ) an m f ∈ M ( B ) , s o as to satisfy the analog of (30 ). If we lo o k at the actio n o f such a n e x tended inner M -map on a g e ne r ic elemen t, namely , the element x ∈ A h x i , and call its image w ( x ) ∈ A h x i , we again ﬁnd that w ( x ) determines the whole extended inner M -map; so we can study suc h maps by examining such elemen ts. (The same observ atio ns apply , with obvious mo diﬁcations, if o ne is interested in as s o ciating to ea ch f : A → B an indexed family o f op era tions on B , each o f a s peciﬁe d arity , satisfying a set of identities relating them with ea ch other and the op erations of B . Then each op era tio n of ar it y n in the family would hav e a generic instance in A h x 1 , . . . , x n i . W e shall say a little more ab out this in § 10 , but will stick to the ca se of a single unary op eration her e .) F o rmalizing, we have Deﬁnition 10. Su pp ose we ar e given a variety V of algebr as in the sense of universal algebr a, and a class of s et -maps m of m emb ers A of V int o themselves, which c onsists of al l set-maps A → A that satisfy a c ertain family of identities in the op er ations of V , and which we c al l “ M -maps”. Then an extended inner M - map of an obje ct A of V wil l me an a function asso ciating t o every obje ct f : A → B of ( A ↓ V ) an M - map m f of B , su ch that for every morphism h : B 1 → B 2 in ( A ↓ V ) , one has (39) m f 2 h = h m f 1 . Clearly the co nc e pt o f an inner endomorphism o f a n ob ject of a g eneral categor y C given by Deﬁnition 4, when r estricted to the case where C is a v ariety V of algebra s, agr ees with the ab ov e deﬁnition. (The inner auto morphisms o f an ob ject of a v ariety are then the inner endo mo rphisms ( β f ) s uch tha t all β f are inv ertible.) O n the other ha nd, sy stems of maps as in the hypo thesis of Theor em 8 can now, as des ired, be describ ed as the extende d inner derivations of our a sso ciative algebra R . In the next s ection we shall similarly consider e x tended inner deriv ations of Lie algebr as. Digression: Having ta lked ab out several versions of the co ncept of a deriv ation, let me for co mpleteness recall tw o more, though I will not discuss “extended inner” versions of these. The most genera l of the versions ment io ned ab ov e , that o f an ( h, h ′ )-deriv ation, is gener alized further by the concept of a deriv ation fro m a K -algebra S to a n ( S, S )-bimo dule B , fo r mally deﬁned by o ur origina l formulas (25), (26) a nd (27). In the last o f these for m ula s, a nd in the analog o f (28) deﬁning the co ncept of an inner deriv ation S → B , the “mu ltiplica tion” on the right-hand sides of these equations is taken to be that of the bimodule structure. (Thus, (37) and (38) a re the cases of (27) and (28) where S 2 is made an ( S 1 , S 1 )-bimo dule by letting S 1 act on the left via the ima ges o f its elements under h, and on the right via their images under h ′ . ) Such a deriv ation is equiv alent to a homomor phism S → S ⊕ B , where S ⊕ B is made a K -algebr a under a multiplication based on the multiplication of S, the bimo dule structure o f B , and the trivia l int e rnal m ultiplica tio n of B . Ea ch inner deriv ation S → B corr esp onds to conjugation by a unit o f the for m 1 + b ( b ∈ B ) . Finally , in gro up theor y , o ne sometimes sp eaks of a le ft or right deriv ation d from a g roup G to a gro up N on which G acts b y automo r phisms. If we write the ac tio n o f G on N a s left sup erscripts in the cas e 14 GEORG E M. BERGMAN of left der iv ations, and as r ight sup ers c r ipts in the ca se of right deriv ations (r equiring it to b e a left actio n in the former cas e and r ight actio n in the latter), and denote the group op eration of N by “ · ” (to avoid confusion a s to which ele men ts such sup erscr ipts are a ttached to), then the iden tities characterizing these t wo so rts of maps are (40) d ( a b ) = d ( a ) · a d ( b ) , resp ectively , d ( a b ) = d ( a ) b · d ( b ) . In the specia l c ase wher e N is a belia n, this conc e pt of deriv atio n ca n b e reduced to the preceding o ne. Indeed, note that a left or right action of G on N is equiv alent to a structure on N of left or right mo dule ov er the g roup ring Z G. T o supply an action on the other side, and so make N a ( Z G, Z G )-bimo dule, we map Z G to Z by the augmentation map, then use the unique actio n of Z on any ab elian group. A deriv ation G → N in the sense of (40) (wr itten now with “ + ” instead of “ · ”) is then a der iv ation from the ring R = Z G to its bimo dule N . Returning to ring theory , let me no te yet another wa y “inner” has b een used, p ossibly rela ted to tha t o f this note. Automor phisms and deriv ations of a K -algebra R ar e actions on R o f cer tain Hopf algebra s, and studen ts of Hopf a lgebras have deﬁned what it mea ns for an action of an ar bitr ary Hopf alg ebra on an algebra to b e inner [8] [22] [26]. I am out of my depth in this situation, and do no t know how close that c o ncept is to the concepts of inner automor phisms and deriv ations deﬁned here; but it app ears to me that it would b e diﬃcult to embrace under the action o f a single Hopf algebra (or bialgebra ) the clas s of constructions (19) with n ra nging ov er all p os itiv e integers. As no ted in [8], another case of an inner action of a Hopf algebra on an algebr a R gives us a concept of an inner gr ading of R b y a given gr oup or monoid. It would b e interesting to explo re the concept of a n “extended inner grading” . 8. In ner deriv a tions an d inner endomorphisms of Lie algebras. Let K b e a c o mm uta tiv e ring, and Lie K the v ariety of Lie alge bras over K. Deriv ations d : L → L are deﬁned for Lie algebr as a s for ass o ciative algebras , with (25) and (26) unchanged, and with Lie brackets replacing multiplication in the a nalog of (27): (41) d ([ r , s ]) = [ d ( r ) , s ] + [ r, d ( s )] . The deriv ations of any Lie algebr a L or asso cia tive algebr a R themselves form a Lie alg e br a under c ommutator br ackets . F or L a Lie algebr a, there is a natural homomorphism from L to its Lie a lgebra o f deriv ations, called the adjoint map , taking each s ∈ L to the der iv ation ad s : L → L given by (42) ad s ( t ) = [ s, t ] ( t ∈ L ) . F or each s, ad s is calle d the inner derivation of L deter mined by s. Clearly , each s ∈ L induces in this way an extende d inner derivation of L in the sense o f Deﬁnition 10 . T o investigate whether these ar e the only extended inner deriv ations, let B : Ring 1 K → Lie K be the functor that sends eac h asso ciative K - algebra R to the Lie a lgebra ha ving the same underlying K -mo dule as R, and Lie br a ck ets given by commutator brack ets, (43) [ s, t ] = s t − t s. W e r ecall that the functor B has a left adjoint, the universal enveloping algebr a functor E : Lie K → Ring 1 K . (The Poincar ´ e- Birkhoﬀ-Witt Theo r em tells us , inter alia, that if L is a Lie algebr a over a ﬁeld, then the natural map L → B ( E ( L )) is an embedding .) Now supp ose ( d f ) is an extended inner deriv ation of L. The adjointness relation b etw een B a nd E tells us that for S an as so ciative K - algebra, homomo r phisms E ( L ) → S as asso ciative algebr as cor resp ond to Lie homomorphisms L → B ( S ); hence if we apply our ex tended inner deriv atio n to the latter homomo rphisms, we get for every ob ject f : E ( L ) → S o f ( E ( L ) ↓ Ring 1 K ) a deriv ation of the Lie algebra B ( S ) , in a functorial manner. The co ndition of be ing a deriv ation of B ( S ) as a L ie algebra is w e aker than that of b eing a der iv atio n of S a s an a sso ciative algebr a, so we can’t apply Theor em 8 dire c tly to this family of deriv ations. The family will, how ever, by the same a rguments as b efor e, b e determined by an element w ( x ) ∈ E ( L ) h x i ; and will satisfy (25) and (26), which we hav e s een are equiv alent to w ( x ) having the for m P a i x b i , with a i , b i now taken from E ( L ) . The diﬀerence b et ween the ass o ciative ca se and the Lie case rear s its head in INNER ENDOMORPHISM S 15 the equation saying that our induced maps sa tisfy (41). This inv o lves co mm uta to r brack ets in E ( L ) h x 0 , x 1 i in pla ce of its asso cia tive multiplication; thus, instea d of (32) we get (44) P n 1 a i ( x 0 x 1 − x 1 x 0 ) b i = (( P n 1 a i x 0 b i ) x 1 − x 1 ( P n 1 a i x 0 b i )) + ( x 0 ( P n 1 a i x 1 b i ) − ( P n 1 a i x 1 b i ) x 0 ) . The added co mplex it y is illusory , how ever! W riting R for E ( L ) , note that the ter ms of (44) lie in the direct sum of tw o comp onents R x 0 R x 1 R ⊕ R x 1 R x 0 R ⊆ R h x 0 , x 1 i . If we pro ject (44) o nt o the ﬁrst of these, we get pre c isely (32 ), and we can rep eat the computations that led us to Theor em 8 . A left inverse to the canonica l map K → E ( L ) , as nee de d for the pro o f of that theorem, is supplied by the algebra homomorphism (45) E ( L ) → K that we get on applying E to the tr ivial map L → { 0 } . What those computations now tell us is that there is a b ∈ E ( L ) such that w ( x ) = x b − b x, so that for an ob ject f : E ( L ) → S of ( E ( L ) ↓ Ring 1 k ) , the induced deriv ation on B ( S ) is given by the op er ation of commutator brack et with f ( b ) . (This shows, inciden tally , that that deriv ation of the Lie algebr a B ( S ) is in fact a deriv ation of the asso ciative algebra S. ) If we now assume that K is a ﬁeld, so that every Lie algebra M can be identiﬁed with its image in E ( M ) , we see that g iven any Lie algebra homo morphism f : L → M , the resulting deriv ation d f : M → M can b e des crib ed within E ( M ) a s commutator bra ck ets with f ( b ) . (Here we are using the fact that by the functoria lity of our extended inner deriv ation, its be havior on M is the restr ic tion of its b ehavior o n B ( E ( M )) . ) Also, since elements o f K induce the z e ro der iv ation, we can assume without lo ss o f genera lit y that the c onstant term o f b (its image under (45 )) is zer o. This r educes our problem to the question: what elements b ∈ E ( L ) with co nstant term 0 hav e the prop erty that fo r ev er y f : L → M , the op eratio n o f commutator bra ck ets with the image o f b in E ( M ) carries M ⊆ E ( M ) in to itse lf ? Equiv alently , what elements b with consta n t term zero have the prop erty that the element w ( x ) = x b − b x ∈ E ( L ) h x i lie s in the Lie subalgebra of E ( L ) h x i genera ted by L and x ? Clearly , all b ∈ L hav e this prop erty . Are they the only o nes? If the ﬁeld K has p ositive characteristic p, the a nswer is no. It is known that in this case the p -th power of a deriv ation o f a Lie or a sso ciative algebra is again a deriv ation, and in particula r, that the p -th power of the inner deriv ation o f an a sso ciative a lg ebra determined by a n element a is the inner deriv ation determined by a p . (This, despite the fac t that the p -th p ow er map do es not in g eneral resp ect addition on noncommutativ e K -a lgebras.) F or nonz e ro a ∈ L ⊆ E ( L ) , the element a p ∈ E ( L ) will not lie in L ; so commutator brack ets with such elements give ex tended inner der iv atio ns of L that do not co me from inner deriv ations in the tr aditional sense. The a bovemen tioned fact ab out p -th p ow er s of deriv ations in characteristic p leads to the concept of a p -Lie algebr a (or r est ricte d Lie a lgebra of characteristic p [1 7, § V.7]): a Lie alge bra L ov er a ﬁeld of characteristic p with a n additional op er a tion o f “ formal p - th power”, a 7→ a [ p ] , satisfying appropria te ident ities . F or this class of ob jects, one has a “restricted universal env elo ping alg ebra” co nstruction, E p ( L ) , where r e lations are imp osed making the fo rmal p -th powers of e le men ts in the p -Lie a lgebra coincide with their ordinar y p -th p ow ers in the enveloping alg e bra. As we shall note below, this lea ds to a mo diﬁed v er sion in character istic p of the question whose unmo diﬁed form we just answered in the negative. When K has characteris tic 0 , I suspec t that a Lie alg ebra L has no extended inner deriv ations o ther than tho s e induced by elements b ∈ L. (If ther e exis ted ge ne r al cons tructions in this cas e, like the p -th power op erator in the characteristic- p c ase, one would expect the phenomenon to b e w e ll-known!) In an y case, we ask Question 11. If L is a Lie algebr a over a ﬁeld of char acteristic 0 , c an c ommutator br ackets with elements b ∈ E ( L ) of c onstant term zer o, other than elements of L, induc e ext ende d inner derivations of L ? Same qu estion for L a p -Lie algebr a over a ﬁeld of char acteristic p > 0 , and b ∈ E p ( L ) . These ar e e quivalent to the questions of whether ther e c an exist in E ( L ) ( r esp e ctively in E p ( L )) elements b of c onstant term zer o not lying in L, with the pr op erty t hat the element w ( x ) = x b − b x b elongs to the Lie sub algebr a ( r esp e ctively the p -Lie su b algebr a ) of E ( L ) h x i ( r esp e ctively E p ( L ) h x i ) gener ate d by L and x. Just as we have us ed, ab ov e, o ur analysis of extended inner der iv atio ns on asso ciative algebr as in studying extended inner der iv ations on Lie algebras , so we can do the same fo r e xtended inner endomorphis ms of 16 GEORG E M. BERGMAN Lie alg ebras, again assuming K a ﬁeld. If we copy the developmen t of Theo rem 6, ta king R = E ( L ) , and using commutator brack ets in place of pro ducts, we c an again get from an extended inner endomor phism of a Lie algebr a L an element w ( x ) ∈ E ( L ) h x i , which we ﬁnd will hav e the form P a i x b i for a i , b i ∈ E ( L ); and the map it induces will resp ect commutator brack ets on ob jects of ( E ( L ) ↓ Ring 1 K ) . That prop erty is eq uiv alent to a formula like (15), but with c ompo nent s in b oth R x 0 R x 1 R a nd R x 1 R x 0 R. Ag a in, pro jection o nt o the R x 0 R x 1 R comp onent gives us precisely o ur old formu la , in this case (15). As in § 2 , this yields (17 ). How ever, homomorphisms o f Lie alg ebras satisfy no a na log of the condition of sending 1 to 1; so we do not ha ve (12), and cannot deduce (13). What does (17) tell us without (1 3)? It says that the ident ity endomorphism of the free rig h t E ( L )-mo dule o f dimension n factors through the free r ight E ( L )-mo dule of dimension 1 . Now E ( L ) admits a homo mo rphism to the ﬁeld K , namely (45 ), so such a factoriza tion of maps of free mo dules can only exist if s uc h a fa ctorization e xists for mo dules ov er K , i.e., if n ≤ 1 . If n = 0 then w ( x ) = 0 , a nd in cont r ast to the cas e of unital asso c ia tive rings, this indeed corresp onds to an inner endomorphism of L in Li e K . If n = 1 , then (17) b ecomes b 1 a 1 = 1 . F r om the fact that the K -a lgebra E ( L ) has a ﬁltratio n whose asso ciated gra ded r ing is a p olynomial ring ov er K , it follows tha t, like a po lynomial ring, it has no 1- sided in vertible elements other than the no nzero elements of K ; so a 1 , b 1 ∈ K, and we co nclude that w ( x ) = x. Hence, Theorem 12. If L is a Lie algebr a over a ﬁeld K , then its only extende d inner endomorphisms ar e the zer o endomorphism and the identity automorphism.  The ab ov e r esult, e ven in the characteristic- p case , concerns ordina r y Lie alg ebras, not p - L ie alg ebras. If K is a ﬁeld o f characteristic p > 0 , and L a p -Lie algebra ov er K , we can b egin the ana lysis of extended inner endomorphisms of L a s a bove, with E p ( L ) in place o f E ( L ) , and go through muc h the same arg ument , us ing as b efore the fact that w ( x ) resp ects Lie br ack ets, and conclude that every e x tended inner endomor phism is either zero, o r induced b y an element w ( x ) = a x b for a, b ∈ E p ( L ) s atisfying b a = 1 . (Note that this automatically implies that w ( x p ) = w ( x ) p . ) B ut w e can no long er say that the relatio n ba = 1 implies that a, b ∈ K. F or example, if u is an element of L such that u [ p ] = 0 , then in E p ( L ) we hav e u p = 0 , so 1 − u is a no nscalar inv ertible element. Hence we ask Question 13. Can a p -Lie algebr a L over a ﬁeld K have a nonzer o non- identity ext ende d inner endomor- phism? Equivalently, c an E p ( L ) have elements a , b , n ot in K , satisfying b a = 1 , and su ch that in E ( L ) h x i , the element w ( x ) = a x b lies in the p -Lie sub algebr a gener ate d by L and x ? The ﬁrs t par t of the a b ove question can b e divided in tw o: Can such an L hav e an extended inner automorphism that is not the identit y? a nd can it hav e a nonzero extended inner endomor phism that is not an extended inner automorphis m? The latter p oss ibility can in turn b e divided in tw o: There mig h t be an inv e rtible element, co njugation by whic h carries the p -Lie s ubalgebra genera ted by L and x into, but not onto, itself, or the extended inner endo morphism might arise from elements a, b such that ba = 1 but ab 6 = 1; I do not know whether an env eloping a lgebra E p ( L ) can contain one-sided but not tw o -sided inv ertible e le men ts. How ever, we can again say that a nonzero extended inner endomorphis m is everywhere one-to- one. F or if our w ( x ) = a x b, and if o n mapping x to so me element u ∈ L ′ under a map L h x i → L ′ , we get a u b = 0 , then by multiplying this equa tion on the left by b a nd on the right by a, we ﬁnd that u = 0 . It is natural to ask whether the methods we hav e used to study inner automorphisms, inner endomor- phisms, and inner deriv ations of a s so ciative and Lie algebras are a pplicable to o ther classes o f not-necessar ily- asso ciative a lgebras. Our results for a sso ciative alg ebras us ed the descr iptions (8) and (9) of the free ex- tensions R h x i and R h x 0 , x 1 i of an algebr a R ; a nd our partial results for Lie alg ebras were base d on reduction to the asso ciative cas e . F or mos t v arieties o f K -alg e bras, the descriptions of the univ e rsal one- and t wo-element ex tensions of an algebra ar e no t so simple. I have no t exa mined what can b e proved in suc h cases. INNER ENDOMORPHISM S 17 9. Co-inner endomorphisms. If A is an ob ject of a c a tegory C , ther e is a constr uction dua l to that o f ( A ↓ C ) , namely ( C ↓ A ) , the ca tegory whose o b jects are o b jects of C given with maps to A, and mo rphisms making commutin g triangles with those maps. Thus, we may dualize Deﬁnition 4 , and deﬁne a n extende d c o-inner endomor phism of an ob ject A o f C to mean a n e ndomorphism E of the forgetful functor ( C ↓ A ) → C , and a c o-inner endomorphism of A itself to mean the v alue o f such a morphism on A. I don’t know of impor ta n t natur ally o ccur r ing exa mples, a nd I susp ect that if the concept turns out to be useful, it will be so mainly in areas other tha n algebra; but let us make a few observ ations on the algebr a case. Let V b e a v ariety of alg ebras in the sense of univ er sal algebra . W e b egin with the weak er concept of an extended co-inner set -map of A ; that is, an e ndo morphism E of the comp osite of forgetful functors (46) ( V ↓ A ) − → V − → Se t . T o a nalyze such a mapping, let us, fo r each a ∈ A, consider the o b ject o f ( V ↓ A ) given by the homomorphism fro m the free V -ob ject on o ne gener ator, h x i V , to A, that takes x to a. If we a pply our co-inner set- ma p E to this homomo r phism, we g et a set-ma p h x i V → h x i V ; this will take x to some element w a ( x ) ∈ h x i V ; thus we get a family o f such ele ments w a ( x ) ∈ h x i V , one for ea ch a ∈ A. W e see that this family will determine E ; namely , for every ob ject f : B → A of ( V ↓ A ) , and every ele men t b ∈ B , E f will take b to w f ( b ) ( b ) . Clearly , any A -tuple ( w a ( x )) a ∈ A of elements of h x i V yields such a n e xtended “co-inner set map” E . (Remark: though there is an added co mplexit y r elative to the case of an ex tended inner endomorphism of an algebr a , in that we now hav e a family of elements w a ( x ) rather than a sing le element w ( x ) , there is a c o rresp onding decrea se in co mplex it y , in that these lie in h x i , r ather than A h x i . ) F or most V , few extended co-inner set maps will give endomor phisms of the alg ebras B . O ne wa y to g et examples which do so is to take all w a ( x ) the same, with v alue g iving what we called in § 5 an “absolute endomorphism of V ”. E.g ., fo r V = Ab and A any a be lian group, we may ta ke all w a ( x ) e qual to n x for a ﬁxed n. More genera lly , for V the v ariet y of mo dules ov er a ring R and A any such mo dule, we may take all w a ( x ) equa l to c x for a ﬁxed element c of the center of R . How ever, here is a class o f cases in which not all c o-inner endomorphisms are based on absolute endomor- phisms. Theorem 14. L et G b e a gr oup, let Set G b e t he variety of right G -sets, let A b e an obje ct of this variety, and let S b e a set of r epr esentatives of t he orbits of A under G. Then every extende d c o-inner endomorphism E of A in Set G is an exten de d c o-inner automorphism , and may b e c onst r u cte d by cho osing, for e ach s ∈ S, an element g s of the c entr alizer in G of the stabilizer G s of s , and for e ach s h ∈ A ( s ∈ S, h ∈ G ) , letting w sh ( x ) = x h − 1 g s h. The ex tende d c o-inner endomorphisms of A thus form a gr oup, isomorphic t o the dir e ct pr o duct, over s ∈ S, of t he c entra lizers of the st abilizer sub gr oups G s . Pr o of. T o get an extended co-inner endomo r phism of A, we m ust choos e for each a ∈ A a n element w a ( x ) of the free G - s et o n one gener ator, which we will deno te x G, in a way that makes the resulting extended inner set-map co nsist of morphisms of G -sets. By the str uc tur e of x G, we see that for ea ch a ∈ A we have w a ( x ) = x g a for a unique g a ∈ G. The condition for thes e maps to induce morphisms of G -s e ts is that for every a ∈ A and h ∈ G, w a ( x ) h = w ah ( x h ) , in o ther words (47) x g a h = ( x h ) g ah . The ab ov e equality is equiv alent to g a h = h g ah , or s olving for g ah , (48) g ah = h − 1 g a h ( a ∈ A, h ∈ G ) . If h lies in the stabilizer s ubgroup G a , we hav e a h = a, so g ah = g a , so in this case (48 ) s ays that g a commutes with h. Hence g a lies in the centralizer of G a . F or general h, (48) allows us to compute g ah from g a , hence, the system of elements g a will be determined by those such that a lies in o ur set of coset representatives S, a nd the v alue at e a ch s ∈ S will b elong to the centralizer of G s . F or elements g s so chosen, it is now easy to verify that we indeed get an extended co-inner endomorphism of A. The resulting endomor phisms are clearly inv ertible, and the description of the gr oup they form is immediate.  18 GEORG E M. BERGMAN 10. Concluding remarks. The to ols used in §§ 1- 8 above are not new fro m the p oint of view of catego ry-theoretic universal a lg ebra. If we consider the g eneral context of “ M -maps ” as in § 7, and then pass to the still more g eneral context, sketc hed parenthetically there, of a family o f additional o pe r ations o n the under lying set of a n ob ject of V , o f v arious arities, sub ject to s ome set of ident ities , we see that this constitutes a structure o f algebra in a v ariety W who se op era tions and iden tities include those of V . An “extended inner system” of such op erations on an o b ject A of V then means a factorizatio n o f the for getful functor ( A ↓ V ) → V thro ugh the forg e tful functor W → V . F rom the p oint of view of the theory o f representable algebra- v alued functors ([15], [5, Chapter 9], [6, Chapter s I-I I]), this cor resp onds to starting with the represe nting ob ject for the former forgetful functor, namely , A h x i with the ca nonical system of co- op erations that make it a co- V - ob ject of the v ariety ( A ↓ V ) , and enhancing that co- V -structure in an arbitra ry way to a co- W -structure; i.e., supplying additional co-op era tions which co-satisfy the identities of W . Thes e co-o p er ations will be determined by their actions on the element x, so by studying the images of x under them, one may attempt to determine the for m that the additional co-o p er ations can take. Thu s , wha t we hav e b een do ing falls under the general study of representable functors and the coa lgebras that represent them. I consider the contribution of this note not to lie in the ma ximu m gener a lit y to which the concepts could b e pushed (which comes to that existing gener al theory), but, inv er sely , in the fo cus on a sp eciﬁc class o f such problems: those where an added unary o per ation constitutes a type of additional structure on the ob jects in question that is alr e a dy of interest, e.g., an endomor phism, o r a deriv ation. W e hav e g otten ex a ct des c r iptions of the po ssibilities for this structure in s everal suc h case s , and shown the techn iq ue tha t can b e applied to further cases. Of cours e , if this note le a ds some readers to an interest in the gener al theory o f coa lgebras and representable functors a mong v ar ieties of algebra s [15], [5, C ha pter 9], [6], I will b e all the mor e pleas e d. 11. Appendix: Inner endomorphisms of associa tive algebras are one-to-one. W e no ted in the second para graph after Theo rem 6 that the o ne-one-ness of every inner endomorphism of an asso cia tive unital alg ebra R ov er a ﬁeld K , which follows from that theorem, also ha s an elementary pro of, using the fact that R can b e em b edded in a s imple K -algebra . W e prov e b elow a diﬀerent em b edding result, from which we deduce, more genera lly , the one-one- ness of all inner endomorphisms o f asso cia tiv e unital algebra s over a rbitrary K. Below, K is, a s usual, a c ommu ta tiv e ass o c iative unital r ing, and ⊗ denotes ⊗ K . K -alg ebras ar e here understo o d to b e as so ciative and unital. Lemma 15. Every K -algebr a R admits an emb e dding f : R → S in a K - algebr a S with the pr op erty t hat for every nonzer o r ∈ R, t he ide al S f ( r ) S c ont ains a nonzer o element of t he c enter of S. Pr o of. Given R , ﬁr st form the K -mo dule R ⊗ R , and no te that the tw o maps R → R ⊗ R given by r 7→ r ⊗ 1 and r 7→ 1 ⊗ r are one-to -one, since the map R ⊗ R → R induced by the internal m ultiplicatio n of R gives a left inv er se to each of them. Now regar d R ⊗ R as a K -alg ebra in the usual way , i.e., so that ( r 1 ⊗ r 2 ) · ( r ′ 1 ⊗ r ′ 2 ) = ( r 1 r ′ 1 ⊗ r 2 r ′ 2 ) . By the ab ov e obser v ation, r 7→ r ⊗ 1 and r 7→ 1 ⊗ r are embeddings of K -algebra s. Note that their images c en tr alize one another, and that the map θ : R ⊗ R → R ⊗ R deﬁned by θ ( r 1 ⊗ r 2 ) = r 2 ⊗ r 1 is an a utomorphism o f R ⊗ R. Using this a utomorphism, let us form the twisted p olynomial algebra ( R ⊗ R )[ t ; θ ]; i.e., adjoin to R ⊗ R an indeterminate t s atisfying (49) t ( r 1 ⊗ r 2 ) = ( r 2 ⊗ r 1 ) t for r 1 , r 2 ∈ R. Within ( R ⊗ R )[ t ; θ ] , we no w take the subalgebr a of ele ments whose consta n t terms lie in R ⊗ 1 , and let S b e the quotient of this subalge bra by the ideal of all elements in which t app ears with expo nen t > 2 . Thu s , as a K -mo dule, (50) S = ( R ⊗ 1) ⊕ ( R ⊗ R ) t ⊕ ( R ⊗ R ) t 2 . W e now deﬁne our algebr a embedding f : R → S by (51) f ( r ) = r ⊗ 1 . INNER ENDOMORPHISM S 19 F or every nonzero r ∈ R, the ideal S f ( r ) S co ntains the element (52) t f ( r ) t = t ( r ⊗ 1) t = (1 ⊗ r ) t 2 , which we see from the rig h t-ha nd side of the ab ov e eq uation is nonzero. Beca use this element inv o lves t to the s econd p ower, it annihilates on b oth sides the summands of (50) inv o lv ing t. It also centralizes the summand R ⊗ 1 , since the factors 1 ⊗ r and t 2 bo th do so. So (52) gives the des ir ed nonzero central element.  T o make use of this result, rec a ll that in our categ ory of K -algebra s, a n endomor phism o f an ob ject by deﬁnition ﬁxes the unit, a nd that in § 2 we transla ted this to the condition (13 ) o n extended inner endomorphisms. The pro of of the next res ult shows that in this r esp e ct, extended inner endomorphisms cannot tell the diﬀerence betw e en the unit and other R -ce ntralizing elements. Lemma 16. If R is a K -algebr a, and ( β f ) an ex t ende d inner endomorphism of R, then for every homo- morphism f : R → S of K - algebr as, the endomorphi s m β f of S ﬁxes al l elements of S that c en tr alize f ( R ) , henc e, in p articular, al l elements of the c enter of S. Pr o of. By abuse of notation, let us use the same symbols for elements of R and their imag es in S. If c ∈ S centralizes R , then applying (11) to c, and co mmuting c past the co eﬃcients b i ∈ R , we get β f ( c ) = P n 1 a i b i c, which by (13) simpliﬁes to c.  W e can now prov e Prop ositio n 1 7. Every inner endomorphism α of an asso ciative unital K - algebr a R is one-to-one. Pr o of. Given R, take a n embedding R → S as in Lemma 1 5. Th us for any nonzero r ∈ R , S f ( r ) S contains a no nzero central element c. By L e mma 16 , β f ( c ) = c. So (53) 0 6 = c = β f ( c ) ∈ β f ( S f ( r ) S ) ⊆ S β f ( f ( r )) S = S f ( α ( r )) S, so α ( r ) 6 = 0 .  Incident a lly , in Lemma 15, we made our construction satisfy the str ong conclusion that S f ( r ) S have nonzero in ters ection with the cen ter of S, since that seemed o f indep endent interest; but for the pr o o f of Prop ositio n 17, it would hav e suﬃced tha t S f ( r ) S hav e no nzero int e rsection with the centralizer of f ( R ) . 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