Infinitely generated projective modules over pullbacks of rings

Infinitely generated pro jectiv e mo dules o v er pullbac ks of rings Dolors Herb era ∗ Departamen t de Matem` atiques, Univ ersitat Aut` onoma de Barcelona, 08193 Bellaterra (Barcelona), Spain e-mail: dolors@mat.uab.cat P av el P ˇ r ´ ıho da † Charles Univ ersity , F ac ult y of Mathematics and Ph ysics, Departmen t of Algebra, Sokolo vsk´ a 83, 18675 Praha 8, Czec h Republic e-mail: priho da@k arlin.mff.cuni.cz Abstract W e use p u ll backs of rings to realize the submonoids M of ( N 0 ∪ {∞} ) k whic h are the set of solutions of a finite system of linear diophantine inequalities as the monoid of isomorphism classes of countably generated pro jective right R -mo dules ov er a suitable semilocal ring. F or these rings, the b eha vior of coun tably generated pro jectiv e left R -mo dules is determined by the monoid D ( M ) defined b y rev ersing the ineq uali ties determining the monoid M . These tw o monoids are not isomorphic in general. As a consequence of our results w e show that there are semilocal rings such that all its pro jectiv e right mo dules are free but t his fails for pro jective left modules. This answers in the n eg ative a question posed by F uller and S hutters [9 ]. W e also provide a ric h v ariety of ex amples of semilocal rings having non fin itely generated pro jectiv e mo dules that are finitely generated mo dulo the Jacobson radical. After the pap er of Ba ss [2] there seemed to b e the g eneral be lie f that the theory of infinitely genera ted pro jective mo dules invite d little inter est . How ever some of the develop- men ts in the re pr esen tation theory of finite dimensio nal a lg ebras [18] and subsequent ones ∗ The final v ersion of this paper w as written while the author w as visi ting NTNU (T ondheim, Norwa y), she thats her host for the kind hospitality . She was also partially supp orted b y M EC-DGESIC (Spain) through Pro ject MTM2008–0620 1-C02-01, and by the Comissionat Per Universitats i Recerca de la Gene ralitat de Catalun y a through Pro ject 2005SGR0020 6. † Supported b y GA ˇ CR 201/09/0816 and research pro ject MSM 002162083 9. 1 in integral repr esen tation theory ha ve drawn the attention to the infinite dimensional rep- resentations [19], [3]. Also the study of the direct sum decomp osition of infinite direct sums of mo dules ov er general rings r equires a go od knowledge of the b eha vior of all pro jectiv e mo dules [17]. As a res ult of this pressur e, interesting general theory on pro jective mo dules has recently app eared [14], [15] and it ha s b een shown that examples of r ings such that not all pro jective mo dules are direct sum of finitely genera ted a re r elativ ely freq ue nt [1 6] and the b ehavior can b e quite complex even for no etherian rings [6]. In this pa per we contin ue this line of work by providing fur ther ex amples of such rings . All of them are semilo cal rings, that is, rings that are s e misimple ar tinia n mo dulo the Jaco bson radical. Our study makes essential use of the result pr o ved by P . Pˇ r ´ ıho da in [14] that, ov er an ar- bitrary ring, pro jectiv e mo dules ar e isomorphic if and only if they are iso morphic mo dulo the Jacobso n radical. F or a semilo cal ring R this implies that the mono id of isomorphism classe s of countably generated pro jectiv e r ig h t (or left) R -mo dules can b e seen as a submonoid of ( N 0 ∪ {∞} ) k for a suitable k ≥ 1, cf. § 1 for the precise definitions. In [10], we characterized the class of monoids that can b e r ealized a s monoid of isomor - phism classes of countably g enerated pro jective right (or left) mo dules ov er a no etherian semilo cal ring as essent i al ly the set of so lutions in N 0 ∪ {∞} of finite homogeneous sy s tems of dio phan tine linear eq uations. In Theorem 1.6 we show that any monoid M which is the set of solutions in N 0 ∪ {∞} of a finite homog eneous system of diopha n tine linear ineq ua li- ties can a ls o b e realiz ed as monoid of isomo rphism classes of countably generated pro jectiv e right mo dules over a suitable s e milocal ring R . In the examples we construct, the monoid of isomorphism classes of countably generated pro jective left R -mo dules is the se t of so lutions in N 0 ∪ {∞} of the sy s tem obtained by reversing the inequalities of the system defining M . While in the no etherian ca se the monoid o f countably genera ted pr o jectiv e right mo dules is isomo r phic to the one of co un tably g enerated pro jective left mo dules , as w e show in this pap er, this is no longer tr ue for g e ne r al semilo cal rings. In this pap er we e mp hasize in the study of pr o jectiv e mo dules that are not finitely generated but that they are finitely ge ne r ated mo dulo the Jaco bson radical. The first example of this kind was provided by Gerasimov and Sakhaev in [5], and the constr uction was further developed by Sakhaev in [2 0]. O ther exa mples appear when studying the direct sum decomp osition o f infinite direct sums of uniserial mo dules [17], [7] and [14]. F rom thes e examples it seemed that the existence of such pro jectiv e modules is rare a nd v ery difficult to handle. With our metho ds we can pro duce a wide v ar ie ty of examples where such pro jectives exist and where their b eha vior is under c ontr ol . In o ur examples, the countably g enerated pro jective mo dules that are finitely genera ted mo dulo the Ja cobson radical, cor respond to the solutions in N 0 of the system of inequalities. Bet ween them we distinguish the finitely generated ones a s the ones that fulfill the eq ua lit y . The techniques we use in this paper are an extension of the ones in [10]. As the title indicates, our ring s a re co nstructed as pullbacks of suitable r ing s, and we take adv antage of [13, Theore ms 2.1, 2.2 and 2.3] in which Milno r describ es al l pro jectiv e mo dules ov er a class of ring pullbacks. A key ing r edien t is the Ge r asimo v-Sakha ev example mentioned above and the co mput ation of its mo noid of is o morphism classes o f countably generated pro jective 2 right (and left) mo dules done in [4]. In § 1 we give an overview of the pap er: we intro duce the monoids of pro jective modules , we define in a pre cise w ay the class of mo noids that we will realize in section 5 a s mono ids of co un tably genera ted pro jective right modules and of co un tably generated pro jective left mo dules o ver suitable s e milo cal rings, a nd we state our main realiza tion Theo rem 1.6. In section 2 we dev elop some theory on pro jectiv e modules that are finitely genera ted mo dulo the Jac o bson r adical which essentially follows [20]. Theorem 2.9 is a slight gener a l- ization of the ma in result in [8]. In section 3 we compute some particular examples to illustrate the co nsequences o f Theorem 1.6. F or ins ta nce, in 3.6, we construct a semilo cal ring such that all pro jective left R -modules ar e free while R has a nonzero (infinitely gener ated) rig ht pro jective mo dule that is not a g e nerator. Suc h an example also shows that the notion of p-c o nnected ring is no t left-right symmetric; this answers in the negative a question in [9, page 3 10]. Reca ll that, following Bas s [2], a ring is (left) p - connected if every nonzero le f t pro jectiv e mo dule is a gener ator. W e also provide examples showing that if R is a se milo cal ring such that R /J ( R ) ∼ = D 1 × D 2 and R has a countably generated, but no t finitely gene r ated, pro jective mo dule that is finitely g enerated mo dulo the Jacobson radical then there is still ro om for countably generated (right a nd left, o r just r igh t) pro jectiv e mo dules that are not direct sums of pro jective mo dules that ar e finitely generated mo dulo the Ja cobson ra dical. This answers in the nega tiv e a ques tion formulated in [4, page 32 61]. In section 4 we dev elop some prop erties of the monoids defined by inequalities. Finally , in section 5 we prove Theorem 1.6. 1 Preliminaries and o v erview All our ring s a re a ssocia tiv e with 1, and ring mo rphism mea ns unital ring morphism. 1.1 Monoids of pro jectiv e mo dules Let R b e a ring. Let V ∗ ( R R ) = V ∗ ( R ) ( V ∗ ( R R )) b e the set of is omorphism classes of countably genera ted pro jectiv e righ t (left) R -mo dules. If P a nd Q are countably gener ated pro jective right R -mo dules then the direct s um induces a n addition on V ∗ ( R ) by setting h P i + h Q i = h P ⊕ Q i , so that V ∗ ( R ) is an a dditiv e mo noid. Similar ly , V ∗ ( R R ) is also a n additive monoid. Let V ( R ) b e the set of isomor phism c la sses of finitely genera ted r ig h t (or left) R -mo dules. Again V ( R ) is an a dditiv e monoid, which can b e identified with a submono id of V ∗ ( R ). Since the functor Hom R ( − , R ) induces a dualit y b et ween the c ategory of finitely genera ted pro jective r igh t R -mo dules and the catego ry o f finitely generated pr o jectiv e left R -modules we iden tify V ( R R ) with V ( R ). So that, we also see V ( R ) as a submo no id of V ∗ ( R R ). Another interesting submono id of V ∗ ( R ) is W ( R R ) = W ( R ) which we define as the set of is omorphism cla s ses of c oun tably gener ated pro jectiv e rig h t R -mo dules that are pure submo dules of finitely generated pro jectiv e mo dules. The submonoid of V ∗ ( R R ), W ( R R ) is 3 defined in a similar wa y . Clear ly , V ( R ) ⊆ W ( R ) ⊆ V ∗ ( R ), and V ( R ) ⊆ W ( R R ) ⊆ V ∗ ( R R ). Notice that W ( R ) \ V ( R ) is also a semigro up. Along the pap er w e will find many examples of (se milo cal) rings R with no n tr ivial W ( R ). Now we give a differ en t kind o f ex ample. Example 1.1 [2] L et R denote t he ring of c ontinuous r e al value d functions over the interval [0 , 1] . L et I = { f ∈ R | ther e exists ε > 0 su ch that f ([0 , ε ]) = 0 } then I is a pr oje ctive pur e ide al of R , cf. [7, Example 3.3] or [4, p. 3263]. The no tation W ( R ) is bor ro wed from the C ∗ -algebra w orld, as we think on this monoid as an a lg ebraic analog ue of the Cuntz monoid defined in C ∗ -algebra s. 1.2 The semilo cal case A ring R is said to be semilo cal if mo dulo its Ja cobson r adical J ( R ) is semisimple artinian, that is, R/J ( R ) ∼ = M n 1 ( D 1 ) × · · · × M n k ( D k ) for suitable divis ion rings D 1 , . . . , D k . F or the rest of our discussion we fix a n o n to ring homomor phism ϕ : R → M n 1 ( D 1 ) × · · · × M n k ( D k ) such that Ker ϕ = J ( R ). Let V 1 , . . . , V k denote a fixed o rdered set of re pr esen tatives of the isomorphism classes of simple r igh t R -modules such that End R ( V i ) ∼ = D i . Let us also fix W 1 , . . . , W k , where W i = Hom R ( V i , R/J ( R )) for i = 1 , . . . , k , as an ordered set of represe n tatives of s imple left R -mo dules. If P R is a coun tably gener ated pro jective right R -mo dule then P /P J ( R ) ∼ = V ( I 1 ) 1 ⊕ · · · ⊕ V ( I k ) k and the ca rdinalit y of the sets I 1 , . . . , I k determines the is omorphism c lass o f P / P J ( R ). By [1 4] (cf. Theor em 2.2) pro jectiv e mo dules are deter mined, up to is omorphism, by its quotient mo dulo the Jaco bson radica l. So that, for a semilo cal ring R , to describ e V ∗ ( R ) we only need to reco rd the cardinality of the sets I i for i = 1 , . . . , k . A similar situation holds for pr o jectiv e left R -mo dules. Note that, by Theorem 2.2(i), in the ca se o f semilo cal rings W ( R ) = { h P i ∈ V ∗ ( R ) | P /P J ( R ) is finitely genera ted } . Similarly , for W ( R R ). 1.3 The dimension monoids for semilo cal rings Let N = { 1 , 2 , . . . } a nd N 0 = N ∪ { 0 } . W e also cons ider the mono id N ∗ 0 = N 0 ∪ {∞} with the additio n determined by the addition on N 0 extended by the rule n + ∞ = ∞ + n = ∞ for any n ∈ N ∗ 0 . F ollowing the notation o f § 1.2, if P is a co un tably generated pro jective right R -module such that P /P J ( R ) ∼ = V ( I 1 ) 1 ⊕ · · · ⊕ V ( I k ) k we set dim ϕ ( h P i )) = ( m 1 , . . . , m k ) ∈ ( N ∗ 0 ) k where, fo r i = 1 , . . . , k , m i = | I i | if I i is finite and m i = ∞ if I i is infinite. Therefore dim ϕ : V ∗ ( R ) → ( N ∗ 0 ) k is a monoid morphism. Similarly , we define a monoid mor phism dim ϕ : V ∗ ( R R ) → ( N ∗ 0 ) k . 4 By Theore m 2.2(ii), dim ϕ : V ∗ ( R ) → ( N ∗ 0 ) k and dim ϕ : V ∗ ( R R ) → ( N ∗ 0 ) k are monoid monomorphisms. Note that dim ϕ ( h R i ) = ( n 1 , . . . , n k ) ∈ N k and that dim ϕ ( W ( R )) = N k 0 ∩ dim ϕ ( V ∗ ( R )) while dim ϕ ( W ( R R )) = N k 0 ∩ dim ϕ ( V ∗ ( R R )) . Definition 1.2 A submonoid A of N k 0 is said to b e ful l affine if whenever a , b ∈ A ar e such that a = b + c for some c ∈ N k 0 then c ∈ A . The class of full affine submonoids of N k 0 containing an element ( n 1 , . . . , n k ) ∈ N k is the precise class of monoids that ca n b e r ealized as dim ϕ ( V ( R )) fo r a semilo cal ring R such that dim ϕ ( h R i ) = ( n 1 , . . . , n k ) [6]. The gener al pro blem we are in terested in is determining which s ubmonoids of ( N ∗ 0 ) k can be realized as dimensio n monoids, that is, as dim ϕ ( V ∗ ( R )) fo r a suitable semilo cal ring R . W e do not know the co mplete solution of this problem but in the next definition we single out some clas ses of mo noids that can b e realized as dimension monoids of semilo cal ring. Definition 1.3 L et k ≥ 1 . (i) A submonoid M of ( N ∗ 0 ) k is s aid to b e a monoid de fined b y a sy stem of equations if it is the set of solutions in ( N ∗ 0 ) k of a system of the form D     t 1 . . . t k     ∈     m 1 N ∗ 0 . . . m n N ∗ 0     ( ∗ ) and E 1     t 1 . . . t k     = E 2     t 1 . . . t k     ( ∗∗ ) wher e D ∈ M n × k ( N 0 ) , E 1 , E 2 ∈ M ℓ × k ( N 0 ) , m 1 , . . . , m n ∈ N , m i ≥ 2 for any i ∈ { 1 , . . . , n } and ℓ , n ≥ 0 . (ii) A submonoid M of ( N ∗ 0 ) k is said to b e a monoid defined by a s ystem of inequa li- ties pr ovide d that ther e exist D ∈ M n × k ( N 0 ) , E 1 , E 2 ∈ M ℓ × k ( N 0 ) , ℓ , n ≥ 0 , and m 1 , . . . , m n ∈ N , m i ≥ 2 for any i ∈ { 1 , . . . , n } , su ch that M is the set of solut i ons in ( N ∗ 0 ) k of D     t 1 . . . t k     ∈     m 1 N ∗ 0 . . . m n N ∗ 0     and E 1     t 1 . . . t k     ≥ E 2     t 1 . . . t k     . (iii) If M ≤ ( N ∗ 0 ) k is define d by a syst em of ine qu a lities as in ( ii ) we define its dual monoid D ( M ) as the set of solutions in ( N ∗ 0 ) k of D     t 1 . . . t k     ∈     m 1 N ∗ 0 . . . m n N ∗ 0     and E 1     t 1 . . . t k     ≤ E 2     t 1 . . . t k     Remark 1.4 1) It is imp ortant t o notic e that N ∗ 0 is n o longer a c anc el lative monoid. So that, for ex a mple, the set of solutions in ( N ∗ 0 ) 2 of t h e e quation x = y is not the same as the set of solutions of 2 x = y + x . 5 2) If M is a monoid define d by a system of ine qualities then t h e monoid D ( M ) dep ends on the p art icular system fixe d to define M . F or an e asy example se e Examples 3.6(ii) and (iii). 3) L et A b e a submonoid of N k 0 c ontaining ( n 1 , . . . , n k ) ∈ N k . It was observe d by Ho chster that A is ful l affine if and only if A is the set of solutions in N k 0 of a system of the typ e app e aring in Definition 1.3 (i)(cf. [10, § 6]). In this c ase, the monoid M = A + ∞ · A is a submonoid of ( N ∗ 0 ) k define d by a system of e quations [10 , Cor ol lary 7.9]. 1.4 Realization r esu lts. Main result F or further quoting we r e c all the main r e sult in [10] which characterized the monoids M that can b e realized a s V ∗ ( R ) for a semilo cal no etherian ring R . F or this class of rings a pro jective mo dule that is finitely g e nerated mo dulo J ( R ) m ust b e finitely g enerated so that W ( R ) = V ( R ) (se e , fo r exa mple, Pr oposition 2.7), and also , by [15], V ∗ ( R R ) ∼ = V ∗ ( R ). Theorem 1.5 L et k ∈ N . L et M b e a submonoid of ( N ∗ 0 ) k c ontaining ( n 1 , . . . , n k ) ∈ N k . Then the fol lowing statement s ar e e quivalent: (1) M is a monoid define d by a system of e quations. (2) Ther e exist a no etherian semilo c al ring R , a semisimple ring S = M n 1 ( D 1 ) × · · · × M n k ( D k ) , wher e D 1 , . . . , D k ar e divisi on rings, and an onto ring morphism ϕ : R → S with Ker ϕ = J ( R ) such that dim ϕ V ∗ ( R ) = M . Ther efor e, dim ϕ V ( R ) = M ∩ N k 0 . In the ab ove statemen t , if F denotes a field, R c an b e c onstructe d to b e an F - a lgebr a such that D 1 = · · · = D k = E is a field extension of F . In this pap er we shall pr ove the following realizatio n result Theorem 1.6 L et k ≥ 1 , and let F b e a field. Le t M b e a submonoid of ( N ∗ 0 ) k define d by a system of ine qualities. L et D ( M ) denote its dual monoid. A ssume that M ∩ D ( M ) c ont ai ns an element ( n 1 , . . . , n k ) ∈ N k . Then ther e exist a semilo c al F -algebr a R , a semisimple F - algebr a S = M n 1 ( E ) × · · · × M n k ( E ) , wher e E is a su i table field ext ensio n of F , and an onto morphism of F -algebr as ϕ : R → S with Ker ϕ = J ( R ) satisfying that dim ϕ V ∗ ( R R ) = M and dim ϕ V ∗ ( R R ) = D ( M ) . Mor e over, dim ϕ W ( R R ) = M ∩ N k 0 , dim ϕ W ( R R ) = D ( M ) ∩ N k 0 , and dim ϕ V ( R ) = M ∩ D ( M ) ∩ N k 0 . F or any semilo cal ring V ( R ) is a finitely g enerated monoid, so is V ∗ ( R ) for R no etherian and semilo cal. As we will show in § 4, monoids defined by a system o f inequalities are still finitely g enerated. But, in genera l, we do not know whether a monoid that can be realized as V ∗ ( R ) for so me semilo cal ring R must b e finitely genera ted. 6 2 Pro jectiv e mo du l es, monoids of pro jectiv es and Ja- cobson radical In this section we want to explain the relatio n b et ween W ( R R ) a nd W ( R R ) completing the results in [8]. W e a lso take the opp ortunit y to state in a (too) precis e wa y r esults on lifting maps b et ween pro jective mo dules modulo an ideal contained in the Jaco bs on radica l. Let I b e a tw o- sided idea l of a ring R , let M and N be rig ht R -modules, and le t f : M → N denote a mo dule homomorphism. By the induced homomo rphism f : M / M I → N / N I we mean the map defined by f ( m + M I ) = f ( m ) + N I fo r any m ∈ M . Recall the following w ell known result. Lemma 2.1 L et R b e any ring, and let I ⊆ J ( R ) b e a t wo -side d ide al of R . L et f : P → Q b e a morphism b etwe en finitely gener ate d pr oje ctive right R - mo dules. Then f is an isomorphism if and only if the induc e d homomorphi sm f : P / P I → Q/QI is an isomorphi sm. In contrast, fo r g e neral pro jective mo dules w e hav e. Theorem 2.2 L et R b e any ring, let P and Q b e pr oje ctive right R -mo dules, and let I ⊆ J ( R ) b e a two-side d ide al of R . (i) [8, P ropos itio n 6.1] A mo dule homomorphism f : P → Q is a pur e monomorphism if and only if so is the induc e d map f : P / P I → Q/QI . (ii) [14, T he o rem 2.3 and its pro of] Le t α : P /P I → Q /QI b e an isomorphism of right R/I -mo dules. L et f : P → Q b e a mo dule homomorph ism such that f = α , and let X b e a fin ite subset of P . Then ther e exists an isomorp hism g : P → Q such t h at g = α and g ( x ) = f ( x ) for any x ∈ X . In p articular, P and Q ar e isomorp hic if and only if they ar e isomorphi c mo dulo the Jac obson r adic al. F or further a pplications we note the following corollar y o f The o rem 2 .2 . Corollary 2.3 L et R b e a ring, and let I ⊆ J ( R ) b e a two-side d ide al. L et P b e a c oun tably gener ate d pr oje ct i ve right R -mo dule. L et f : P → P b e a homomorp hism such t ha t the induc e d map f : P /P I → P /P I is the identity, and let X b e a finite su bset of P . Then ther e exists a bije ctive homomorphism h : P → P such t hat the induc e d homomorph ism h = Id P /P I and such that hf ( x ) = x for any x ∈ X . Proof. By Theo r em 2.2(ii), ther e exists an is omorphism g : P → P s uc h that g = Id P /P I and g ( x ) = f ( x ) for a n y x ∈ X . Set h = g − 1 to conclude. Lemma 2.4 L et R b e a ring, let P and Q b e pr oje ctive right R -mo dules. L et I b e a two- side d ide al of R c ontaine d in J ( R ) , and let α : Q/QI → P /P I and β : P /P I → Q/ QI b e homomorph isms su ch that β ◦ α = Id Q/QI . L et f : Q → P and g : P → Q b e mo dule homomorph isms such that f = α and g = β . If f ◦ g is idemp otent then P ∼ = Q ⊕ Q ′ and Q ′ /Q ′ I ∼ = (Id P /P I − αβ )( P / P I ) . 7 Proof. Since f g ( P ) is a dire c t summand of P , f g ( P ) /f g ( P ) I = f g ( P ) / ( f g ( P ) ∩ P I ) ∼ = ( f g ( P ) + P I ) /P I . Since, for any x ∈ P , β ( f g ( x ) + P I ) = β ( x + P I ) we deduce that β : f g ( P ) /f g ( P ) I → Q/ QI is bijective. By Theorem 2.2, w e conclude that Q ∼ = f g ( P ). Since ((Id P − f g )( P ) + P I ) / P I = (Id P /P I − αβ ) ( P /P I ), it follows that Q ′ = (Id P − f g )( P ) ha s the claimed prop erties. Corollary 2.5 L et R b e a ring with Jac obson r adic al J ( R ) . L et I ⊆ J ( R ) b e a two-side d ide al. L et P and Q b e pr oje ctive right R -mo dules su ch that Q is finitely gener ate d. If t h er e exists a pr oje ctive right R /I -mo dule X such that P /P I ∼ = Q/QI ⊕ X then ther e exists a pr oje ctive right R -mo dule Q ′ such that P ∼ = Q ⊕ Q ′ and Q ′ /Q ′ I ∼ = X . Proof. Since Q is finitely genera ted, the split exa ct sequence o f R /I - mo dules 0 → X → P / P I β → Q/ QI → 0 lifts to a split exact sequence 0 → Ker g → P g → Q → 0 where g = β . Therefore P ∼ = Q ⊕ Ker g . W e wan t to show that K er g / (Ker g ) I ∼ = X . Let α : Q /QI → P / P I b e such tha t β α = Id Q/QI , and let f : Q → P b e such that f = α . Since Q is finitely g enerated and g f = β α = Id Q/QI , g f : Q → Q is in vertible (cf. Lemma 2.1). So that, there exists an inv ertible endomorphism h o f Q satisfying that h = Id Q/QI , and such that g ( f h ) = Id. Ther efore, ( f h ) g is an idemp otent endomorphism of P and since (Id − ( f h ) g ) P = Ker g we conclude, by the second part of Le mm a 2.4, that Q ′ = K er g has the claimed pr operties. In the following lemma we reca ll the pro perties o f s equences { f n } n ≥ 1 satisfying that f n +1 f n = f n . Laza rd in [12] realized the imp ortance of them to describ e pure ideals o f a ring. They play a fundamental rˆ ole in constructing finitely generated flat modules ov er semilo cal rings tha t ar e no t pro jective or, equiv alently , in constr ucting non-finitely g e nerated pro jective mo dules tha t a re finitely generated mo dulo the Jacobs on radica l. They w ere v ery w ell analyzed b y Sa khaev in several pap ers, see fo r ex a mple [20]. Re- cently , they hav e been ex tensiv e ly r e-studied [7 ], [8] and [4]. Lemma 2.6 L et R b e any ring. L et P b e a right R -mo dule and let f 1 , . . . , f n , . . . b e a se quenc e of endomorphi sms of P satisfying t h at, for e ach n ≥ 1 , f n +1 f n = f n then, (i) S n ≥ 1 f n · End R ( P ) is a pr oje ctive pur e right ide al of End R ( P ) . (ii) Q = S n ≥ 1 f n ( P ) is a pur e submo dule of P isomorphic to a dir e ct summand of P ( N ) . In p articular, if P is pr oje ctive then so is Q . 8 Proof. ( i ) . This is due to Laza rd [12]. ( ii ). The pur it y of I ins ide S gives I ⊗ S P ֒ → S ⊗ S P . Using the identification S ⊗ S P ≃ P , we get S n ≥ 1 f n ( P ) ≃ I ⊗ S P . Hence the purity of Q inside P follows from the asso ciativity of the tenso r pro duct and ( i ). Consider the co un table direct sy stem P 1 f 1 → P 2 · · · P n f n → P n +1 · · · where P = P n for any n ≥ 1 . Since f n +1 f n = f n , the se q uence { f n } n ≥ 1 induces an injective map f : lim − → P n → P such that Im f = Q . Therefore, Q fits into the (pure) exact s equence 0 → ⊕ n ≥ 1 P n Φ → ⊕ n ≥ 1 P n → Q → 0 where, fo r each n ≥ 1 and letting ε n : P n → ⊕ n ≥ 1 P n denote the ca nonical embedding, the map Φ is deter mined by Φ ε n ( x ) = ε n ( x ) − ε n +1 f n ( x ) for each x ∈ P n . The prop erties of the seq uence of maps { f n } n ≥ 1 imply that Φ s plits see, for ex ample, [1, Prop osition 2.1]. Prop osition 2 . 7 L et R b e a ring. L et P R and Q R b e pr oje ctive right R -mo dules such that P R is finite gener ate d. L et α : Q/Q J ( R ) → P / P J ( R ) and β : P /P J ( R ) → Q/Q J ( R ) b e such that β α = I d Q/QJ ( R ) . L et ε : Q → P b e any mo dule homomorph ism such t hat ε = α . Then ther e exists a se quenc e f 1 , . . . , f n , . . . of endomorphisms of P such t ha t, for e ach n ≥ 1 , f n +1 f n = f n , f n = α ◦ β and Q ∼ = ε ( Q ) = S n ≥ 1 f n ( P ) . Mor e over Q is fi n itely gener ate d if and only if ther e exists n 0 such that f 2 n 0 = f n 0 . In this c ase, f 2 n 0 + k = f n 0 + k for any k ≥ 0 . Proof. Let ϕ : P → Q b e a lifting of β . Note that Q R m ust b e a countably genera ted pro jective mo dule, so that we ca n fix an ascending c hain ∅ = X 1 ⊆ X 2 ⊆ X 3 ⊆ · · · ⊆ X n ⊆ . . . of finite subsets of Q such that X = S n ≥ 1 X n generates Q . Since P is finitely generated and using Coro llary 2.3, we ca n construct, inductiv ely , a sequence Id Q = h 1 , . . . , h n , . . . of (auto)mor phisms o f Q such that if, for each n ≥ 1, we set f n = εh n h n − 1 · · · h 1 ϕ then h n +1 h n · · · h 1 ϕf n = h n · · · h 1 ϕ and h n +1 h n · · · h 1 ϕε ( x ) = x for any x ∈ X n +1 . It can b e ea s ily ch eck ed that the homomorphisms { f n } n ≥ 1 satisfy the desired prop erties. If Q is finitely gener ated there exis ts n 0 such that ε ( Q ) = f n 0 − 1 ( P ). Observe that f n 0 f n 0 − 1 = f n 0 − 1 says f n 0 ( x ) = x fo r any x ∈ Q . In pa r ticular, f 2 n 0 + k = f n 0 + k for any k ∈ N . Conv e rsely , in view of Lemma 2.4, if there exists n 0 such that f 2 n 0 = f n 0 then Q is isomorphic to f n 0 ( P ) which is a direct summand o f P . In particular, Q is finitely g enerated and f n 0 ( P ) = f n 0 + k ( P ) for any k ≥ 0. Since f n 0 is idemp oten t, for any k ≥ 0, f n 0 + k = f n 0 f n 0 + k so that f 2 n 0 + k = f n 0 + k f n 0 f n 0 + k = f n 0 + k . 9 Remark 2.8 In the situation of Pr op osition 2.7, fix n ≥ 1 . N o tic e t h at ( f n +1 − f n ) f n = f n − f 2 n . Sinc e f n +1 − f n = 0 ∈ E nd R ( P /P I ) and P is a finitely gener ate d pr oje ctive mo dule, u = Id P − ( f n +1 − f n ) is a unit such that uf n = f 2 n . F or any m ∈ Z , set g m = u − ( m +1) f n u m ∈ End R ( P ) . It e asily fol lows that, for any m ∈ Z , g m +1 g m = g m and also that (Id P − g m +1 )(Id P − g m ) = Id P − g m +1 so that, by L emma 2.6, P ′ n = S m ≥ 0 g m P is a pr oje ctive pur e su bmo dule of P and Q ′ n = S m ≤ 0 Hom R ( P, R )(Id P − g m ) is a pr oje ct i ve pur e submo dule of t he pr oje ctive left R -mo dule Hom R ( P, R ) . Notic e t h at, for any m , g m = α ◦ β and Id P − g m = Id P /P I − α ◦ β . Ther efor e, P ′ n /P ′ n I ∼ = Q/QI , henc e P ′ n ∼ = Q , and Q ′ n /I Q ′ n ∼ = Hom R/RI ((Id P /P I − α ◦ β ) P /P I , R/I ) . In p articular, t h e isomorphism classes of P ′ n and Q ′ n , r esp e ctively, do n ot dep end on n . Combining Prop osition 2.7 with Remar k 2 .8 w e o btain the fo llo wing theorem w hich is a slight refinemen t of [8, Theor em 7.1 ]. Theorem 2.9 L et R b e a ring, let P b e a fi nitely gener ate d pr oje ctive right R -mo dule, and let I ⊆ J ( R ) b e a t wo -side d ide al of R . Assum e that t h er e is a split exact se quenc e of right R/I mo dules 0 → X → P / P I → X ′ → 0 . Then the fol lowing statement s ar e e quivalent, (i) Ther e exists a (c ountably gener ate d) pr oje ctive right R -mo dule Q su ch t hat Q/QI ∼ = X . (ii) Ther e exists a (c oun tably gener ate d) pr oje ctive left R -mo dule Q ′ such that Q ′ /I Q ′ ∼ = Hom R/I ( X ′ , R/I ) . When the ab ove e quivalent statement hold Q is isomorphic to a pur e submo dule of P , and Q ′ is isomorphic to a pure submo dule of Ho m R ( P, R ) . Mor e over, Q is fi nitely gener ate d if and only if Q ′ is fin it ely gener ate d if and only if ther e exists a pr oje ctive right R -mo dule P ′ such that P ′ /P ′ I ∼ = X ′ . Now we are going to state some of the results ab o ve in ter ms of monoids o f pro jectiv es. More precisely , in terms of pre-order ed mo noids o f pr o jectiv es. W e recall that over a comm utative monoid M there is a pre-o rder relation c a lled the algebr aic pr e or der on M defined by x ≥ y , for x , y ∈ M , if and only if x = y + z for some z ∈ M . F or exa mple, o ver ( N ∗ 0 ) k the algebr a ic order is the comp onent -wise or der, which is even a partial o rder. When the monoid is V ∗ ( R ) for some ring R , h Q i ≤ h P i if and only if Q is isomorphic to a direct summand of P . In terms of monoids of pro jectiv e mo dules Corollar y 2.5 e s sen tia lly says that for elements in V ( R ) the alg ebraic preorder is r espected modulo J ( R ). W e state this in a precise wa y in the next re s ult. 10 Corollary 2.10 L et R b e a ring, and let I b e a two-side d ide al of R c ontaine d in J ( R ) . L et π : R → R/I denote the pr oje ction, and let ∼ π : V ∗ ( R ) → V ∗ ( R/I ) denote the induc e d homomorph ism of monoids. If x ∈ V ∗ ( R ) , y ∈ V ( R ) ar e such that ther e exist c ∈ V ∗ ( R/I ) satisfying that ∼ π ( x ) = ∼ π ( y ) + c then ther e exists z ∈ V ∗ ( R ) such t ha t ∼ π ( z ) = c and x = y + z . In general, for a semilo cal ring R , the monoid V ∗ ( R ) is isomo rphic to a submonoid of ( N ∗ 0 ) k . In view of Theorem 2.2, the alg ebraic order of ( N ∗ 0 ) k induces an order on V ∗ ( R ) that is transla ted in terms of pro jective mo dules ov e r R by h Q i ≤ h P i if a nd o nly if ther e exis ts a pure monomorphism f : Q → P if and only if Q/QJ ( R ) is a direct summand of P /P J ( R ). By [14], the rela tion ≤ is antisymmetric. This pa r tial or der rela tion defined o n V ∗ ( R ) restricts to the usual algebr aic order ov er V ( R ), but not on W ( R ) when V ( R ) ( W ( R ). Corollary 2.11 L et R b e a semilo c al ring, fix ϕ : R → S an onto ring homomorph ism t o a semisimple artinian ring S such that Ker ϕ = J ( R ) . Then (i) x ∈ W ( R ) \ V ( R ) if and only if x is inc omp ar able (with r esp e ct to the algebr aic or der) with n h R i for any n ≥ 1 if and only if ther e exist n ≥ 1 such t h at n · dim ϕ h R i − dim ϕ ( x ) ∈ dim ϕ W ( R R ) \ dim ϕ V ( R ) . (ii) V ( R ) = W ( R ) ∩ W ( R R ) . Proof. Since ov er a semisimple a rtinian ring any exac t sequence splits, the statement follows by applying Theorem 2.9. Remark 2.12 Cor ol lary 2.11 implies that, if dim ϕ V ∗ ( R ) ⊆ ( N ∗ 0 ) k is a monoid define d by ine qualities and dim ϕ ( h R i ) ∈ dim ϕ V ∗ ( R ) ∩ D (dim ϕ V ∗ ( R )) , the elements of t h e semigr oup dim ϕ W ( R ) \ dim ϕ V ( R ) must b e the elements of N k 0 such that some of the ine qualities they satisfy ar e strict. So that dim ϕ V ( R ) = dim ϕ V ∗ ( R ) ∩ D (dim ϕ V ∗ ( R )) ∩ N k 0 = dim ϕ W ( R ) ∩ dim ϕ W ( R R ) . In terms of o rder re la tions on the monoids w e have the following Corollar y . Corollary 2.13 L et R b e a semilo c al ring. Consider t h e fol lowing r elation over V ∗ ( R ) , h P i ≤ h Q i if and only if P /P J ( R ) is isomorphic to a dir e ct summ and of Q/ QJ ( R ) . Then (i) h P i ≤ h Q i if and only if t h er e ex ists a pur e emb e dding f : P → Q . (ii) ≤ is a p artial or der r elation that re fines the algebr aic or der on V ∗ ( R ) . (iii) If, in addition, R is no etherian then the p artial or der induc e d by ≤ over V ∗ ( R ) is the algebr aic or der. 11 Proof. ( i ) . If h P i ≤ h Q i then there ex ists a splitting monomo rphism f : P /P J ( R ) → Q/QJ ( R ) which by Theorem 2.2(i) lifts to a pure monomo rphism f : P → Q . Conv ersely , if f : P → Q is a pure monomo rphism of r igh t R - modules then the induced map f ⊗ R R/J ( R ) : P ⊗ R R/J ( R ) → Q ⊗ R R/J ( R ) is a pure monomor phism of R/ J ( R )-mo dules. Since R/ J ( R ) is semisimple, f ⊗ R R/J ( R ) is a split monomorphism. ( ii ) . It is c lear that ≤ is reflexive a nd trans itiv e. As it is alr eady o bs erv ed in [14], Theorem 2.2 implies that ≤ is also a n tisymmetr ic. If P is iso morphic to a direct summand of Q , then P /P J ( R ) is also isomorphic to a direct summand of Q/QJ ( R ). Hence h P i ≤ h Q i , that is , ≤ refines the algebra ic or der o n V ∗ ( R ). ( iii ) . It is a consequence of the r ealization Theo r em 1.5. W e shall s ee in Ex a mples 3.6 that the mo noid V ∗ ( R ) do es not determine V ∗ ( R R ). The- orem 2.9, or [8, Theorem 7.1], combined with T he o rem 2 .2 (ii) implies that for a semilo cal ring W ( R ) do e s determine W ( R R ). Corollary 2.14 F or i = 1 , 2 , let R i b e a semilo c al ring and let ϕ i : R i → M n 1 ( D i 1 ) × · · · × M n k ( D i k ) b e an onto ring homomorphism s u ch that Ker ϕ i = J ( R i ) and D i 1 , . . . , D i k ar e divisio n rings. Then dim ϕ 1 W ( R 1 ) = dim ϕ 2 W ( R 2 ) if and only if dim ϕ 1 W ( R 1 R 1 ) = dim ϕ 2 W ( R 2 R 2 ) . Proof. By sy mm etry , it is enough to prov e that if dim ϕ 1 W ( R 1 ) = dim ϕ 2 W ( R 2 ) then dim ϕ 1 W ( R 1 R 1 ) ⊆ dim ϕ 2 W ( R 2 R 2 ). Let x ∈ dim ϕ 1 W ( R 1 R 1 ). There exists m ∈ N s uch that x ≤ m ( n 1 , . . . , n k ). By Theorem 2.9, y = m ( n 1 , . . . , n k ) − x ∈ dim ϕ 1 W ( R 1 ) = dim ϕ 2 W ( R 2 ). Applying ag ain Theorem 2.9, we deduce tha t x = m ( n 1 , . . . , n k ) − y ∈ dim ϕ 2 W ( R 2 R 2 ). 3 Some examples Gerasimov and Sakhaev gave the first example of a semilo cal ring such that V ( R ) & W ( R ). The final step for the computation o f V ∗ ( R ) w as made in [4]. W e want to s tart this section stating the main prop erties of this exa mple a s it is one of the basic to ols to prov e our realization Theo rem 1.6. Theorem 3.1 ([5], [4]) L et F b e any fi eld . Ther e exists a semilo c al F -algebr a R with an onto ring morphism ϕ : R → F × F with Ker ϕ = J ( R ) and such that al l finitely gener ate d pr oje ctive mo dules ar e fr e e bu t dim ϕ W ( R R ) = { ( x, y ) ∈ N 0 | x ≥ y } = (1 , 1) N 0 + (1 , 0) N 0 dim ϕ V ∗ ( R R ) = (dim ϕ W ( R R )) N ∗ 0 = { ( x, y ) ∈ N ∗ 0 | x ≥ y } and dim ϕ W ( R R ) = { ( x, y ) ∈ N 0 | y ≥ x } = (1 , 1) N 0 + (0 , 1) N 0 dim ϕ V ∗ ( R R ) = (dim ϕ W ( R R )) N ∗ 0 = { ( x , y ) ∈ N ∗ 0 | y ≥ x } 12 In p articular, any pr oje ctive mo dule over R is a dir e ct sum of inde c omp osable pr oje ctive mo dules that ar e fi nitely gener ate d mo dulo J ( R ) . It is quite a n in teresting question to determine the structure of V ∗ ( R ) for a general semilo cal ring. But right now it seems to b e to o challenging even fo r semilo cal rings R suc h that R/J ( R ) ∼ = D 1 × D 2 where D 1 , D 2 are division ring s. Now we pr o vide some examples of such r ings to illustra te Theorem 1.6 and the difficulties that appear in the g eneral c ase. W e first observe that, since k = 2 and (1 , 1) is the order unit o f dim ϕ V ( R ), to ha ve some ro om for interesting b eha vior of c o un tably generated pro jective mo dules all finitely generated pro jective mo dules must b e free. Lemma 3.2 L et R b e a semilo c al ring such t hat R /J ( R ) ∼ = D 1 × D 2 for suitable divi sion rings D 1 and D 2 . Fix ϕ : R → D 1 × D 2 an onto ring homomorph ism such that Ker ϕ = J ( R ) . If R has non-fr e e finitely gener ate d pr oje ctive right (or left) mo dules then ther e exists n ∈ N such that dim ϕ V ( R ) is the submonoid of N 2 0 gener ate d by (1 , 1) , ( n, 0 ) and (0 , n ) . In this c ase, dim ϕ V ∗ ( R ) = (1 , 1) N ∗ 0 + ( n, 0) N ∗ 0 + (0 , n ) N ∗ 0 = { ( x , y ) ∈ N ∗ 0 | x + ( n − 1) y ∈ n N ∗ 0 } . Ther efor e, al l pr oje ctive mo dules ar e dir e ct su m of finitely gener ate d pr oje ct ive mo dules. Proof. Note that dim ϕ ( h R i ) = (1 , 1). So that (1 , 1) ∈ A = dim ϕ V ( R ). Let P b e a non-free finitely generated pro jective right R -module, and let dim ϕ ( h P i ) = ( x, y ). As P is not free, e ither x > y o r x < y . Ass ume x > y , then ( x, y ) = ( x − y , 0) + y (1 , 1) ∈ A ( ∗ ) . Since, by Coro llary 2.5 or its monoid v ersion Corollar y 2.10, A is a full affine submonoid of N 2 0 we deduce that ( x − y , 0) ∈ A and also that (0 , x − y ) = ( x − y )(1 , 1) − ( x − y , 0) ∈ A . If x < y w e deduce, in a sy mm etric wa y that ( y − x, 0) and (0 , y − x ) are elements of A . Cho ose n ∈ N minimal with resp ect to the prop ert y ( n, 0) ∈ A , and note that then also (0 , n ) ∈ A . W e claim that A = (1 , 1) N 0 + ( n, 0) N 0 + (0 , n ) N 0 . W e o nly need to prove that if ( x, y ) ∈ A then it can b e written a s a linear combination, with co efficien ts in N 0 of (1 , 1), ( n, 0) a nd (0 , n ). In view of the prev ious ar gumen t, it suffices to show that if ( x, 0) ∈ A then ( x, 0) ∈ ( n, 0) N 0 . By the division alg orithm ( x, 0) = ( n, 0) q + ( r, 0 ) with q ∈ N 0 and 0 ≤ r < n . As A is a full affine submonoid of N 2 0 we deduce that ( r, 0) ∈ A . By the cho osing of n , r = 0 as desired. Let P 1 be a finitely gener ated right R -mo dule such tha t dim ϕ ( h P 1 i ) = ( n, 0), and let P 2 be a finitely generated right R -module such that dim ϕ ( h P 2 i ) = (0 , n ). Let Q b e a coun tably generated pro jectiv e r igh t R -mo dule that is not finitely generated. Let dim ϕ ( h Q i ) = ( x, y ) ∈ N ∗ 0 . W e wan t to show that ( x, y ) ∈ (1 , 1) N ∗ 0 + ( n, 0) N ∗ 0 + (0 , n ) N ∗ 0 13 If x = y then ( x, y ) = x (1 , 1) and, b y Theorem 2 .2( ii), Q is free. If x > y then y ∈ N 0 and ( x, y ) = ( x − y , 0) + y (1 , 1), co m bining Theorem 2.2(ii) with Lemma 2.5 we deduce that Q = y R ⊕ Q ′ with Q ′ such that dim ϕ ( h Q ′ i ) = ( z , 0) where z = x − y . If z < ∞ then, by Theorem 2.2(ii), nQ ′ ∼ = z P 1 hence Q ′ , and Q , are finitely ge nerated. If z = ∞ , b y Theorem 2.2(ii), Q ′ ∼ = P ( ω ) 1 . Hence ( x, y ) = ∞ · ( n, 0 ) + y (1 , 1). T he ca se x < y is done in a symmetric way . It is not difficult to chec k that the elemen ts of dim ϕ V ∗ ( R ) a re the solutio ns in N ∗ 0 of x + ( n − 1) y ∈ n N ∗ 0 . Now w e will list a ll the po s sibilities for the mo noid V ∗ ( R ) viewed as a submono id of V ∗ ( R/J ( R )) when R is a noe th erian r ing suc h that R/J ( R ) ∼ = D 1 × D 2 , for D 1 and D 2 division rings, and all finitely generated pr o jective mo dules a re free. In v iew of Theorem 1 .5 this is eq uiv alent to classify the submonoids of ( N ∗ 0 ) 2 containing (1 , 1) and that a re defined by a sys tem of equations. Tho ug h the presentation of the monoid as solutions of equations is quite attractive there is a n a lternativ e one that, ev en b eing tec hnical, is mo r e useful to work with. Definition 3.3 Fix k ∈ N and an or der unit ( n 1 , . . . , n k ) ∈ N k . A system of supp orts S ( n 1 , . . . , n k ) c onsists of a c ol le ct i on S of subsets of { 1 , . . . , k } to gether with a family of c ommutative monoids { A I , I ∈ S } such t hat t h e fol lowing c onditions hold (i) ∅ and { 1 , . . . , k } ar e elements of S . (ii) F or any I ∈ S , A I is a su bmo noid of N { 1 ,...,k }\ I 0 . The monoid A { 1 ,...,k } is the t riv ial monoid and ( n 1 , . . . , n k ) ∈ A ∅ . (iii) S is close d under unions, and if x ∈ A I for some I ∈ S t h en I ∪ s upp ( x ) ∈ S . In p articular { 1 , . . . , k } ∈ S . (iv) Supp ose that I , K ∈ S ar e such that I ⊆ K and let p : N { 1 ,...,k }\ I 0 → N { 1 ,...,k }\ K 0 b e the c anonic al pr oje ction. Then p ( A I ) ⊆ A K . If in addi tion, for any I ∈ S , t h e su bmonoids A I ar e ful l affine su bmo noids of N { 1 ,...,k }\ I 0 then S ( n 1 , . . . , n k ) is said to b e a full a ffine system of supp orts . Remark 3.4 Given a system of su pp orts S ( n 1 , . . . , n k ) = { A I , I ∈ S } we c an asso ciate to it a monoid. Consider the subset M ( S ) of ( N ∗ 0 ) k define d by x ∈ M ( S ) if and only if I = inf - supp ( x ) ∈ S and p I ( x ) ∈ A I , wher e if x = ( x 1 , . . . , x k ) then inf - s upp ( x ) = { i ∈ { 1 , . . . , k } | x i = ∞} , and p I : N k 0 → N { 1 ,...,k }\ I 0 denotes the c anonic al pr oje ction. By [10 , The or em 7.7], S ( n 1 , . . . , n k ) is a ful l affine syst em of supp orts if and only if M ( S ) is a m onoid define d by e quations and c ontaining ( n 1 , . . . , n k ) . W e reca ll that a mo dule is sup erdecomp osable if it ha s no indecomp osable direct sum- mand. By Theorem 1.5 and Lemma 2 .5 , in o ur context sup erdecompo sable modules are 14 relatively frequent as they co rrespo nd to the elements x ∈ M ⊆ ( N ∗ 0 ) k such that, for any y ∈ M ∩ N k 0 , supp ( y ) " supp ( x ). Example 3.5 L et R b e a s emilo c al no etherian ring such that ther e exists ϕ : R → D 1 × D 2 , an onto ring morphism with Ker ϕ = J ( R ) , wher e D 1 and D 2 ar e division rings. Assume that al l finitely gener ate d pr oje ctive right R - mo dules ar e fr e e. Henc e dim ϕ V ( R ) = (1 , 1) N 0 , and its or der u nit is (1 , 1) . Then ther e ar e the fol lowing p ossibilities for dim ϕ V ∗ ( R ) : (0) Al l pr oje ctive m o dules ar e fr e e, so that M 0 = dim ϕ V ∗ ( R ) = (1 , 1) N ∗ 0 . Note that M 0 is the set of solutions ( x, y ) ∈ ( N ∗ 0 ) 2 of t h e e qu a tion x = y . (1) M 1 = dim ϕ V ∗ ( R ) = (1 , 1) N ∗ 0 + (0 , ∞ ) N ∗ 0 . So that, M 1 is the set of solutions ( x, y ) ∈ ( N ∗ 0 ) 2 of t h e e qu a tion x + y = 2 y . Note that for such an R t her e exists a c ountably gener ate d sup er de c omp osable pr oje c- tive right R -mo dule P such that dim ϕ ( h P i ) = (0 , ∞ ) . Then any c ountably gener ate d pr oje ctive right R mo dule Q is isomorphi c to R ( n ) ⊕ P ( m ) for su i table n ∈ N ∗ 0 and m ∈ { 0 , 1 } . (1’) M ′ 1 = dim ϕ V ∗ ( R ) = (1 , 1) N ∗ 0 + ( ∞ , 0 ) N ∗ 0 . So that, M ′ 1 is the set of solutions ( x, y ) ∈ ( N ∗ 0 ) 2 of t h e e qu a tion x + y = 2 x . (2) M 2 = dim ϕ V ∗ ( R ) = (1 , 1) N ∗ 0 + ( ∞ , 0 ) N ∗ 0 + (0 , ∞ ) N ∗ 0 . So that, M 2 is the set of solutions ( x, y ) ∈ ( N ∗ 0 ) 2 of t h e e qu a tion 2 x + y = x + 2 y . Note that for such an R ther e exist two c ountably gener ate d s u p er de c omp osable pr o- je ctive right R -mo dules P 1 and P 2 such that dim ϕ ( h P 1 i ) = (0 , ∞ ) and dim ϕ ( h P 2 i ) = ( ∞ , 0) . Any c ountably gener ate d pr oje ctive right R mo dule Q satisfies that t he r e exist n ∈ N 0 and m 1 , m 2 ∈ { 0 , 1 } such that Q = R ( n ) ⊕ P ( m 1 ) 1 ⊕ P ( m 2 ) 2 . Proof. In view o f Theo rem 1.5 and Remark 3.4 we m ust describ e all the p ossibilities for full affine sys tems of supp orts of { 1 , 2 } such that A ∅ = (1 , 1) N 0 . Since the set of supp o rts of a system of suppo rts a t least con tains ∅ and { 1 , 2 } there are just fo ur p ossibilities. Since the image o f the pro jections of A ∅ on the first and o n the second comp onent is N 0 , all the mo no ids A I in the definition o f system of suppo rts a re deter mined b y A ∅ . Case (0) is the one in which M 0 = A ∅ + ∞ · A ∅ . Acco rding to Remark 1.4 (3), in this case all pro jectiv e mo dules are direct sum of finitely generated (indecomp osable) mo dules. In ca ses (1) and (1 ′ ) there ar e 3 different supp orts for the elements in the monoid, and in case (2) there are 4. Now we giv e some examples who se exis tence is a direct consequence o f Theor em 1.6. Example 3.6 L et F b e any field. In al l the statement s R denotes a semilo c al F-algebr a, and ϕ : R → E × E denotes an ont o ring homomorphism su ch that Ker ϕ = J ( R ) and E is a suitable field extension of F . Fix n ∈ N . Then ther e exist R and ϕ su ch t hat 15 (i) N = dim ϕ V ∗ ( R R ) = (1 , 1) N ∗ 0 +( n, 0) N ∗ 0 = { ( x , y ) ∈ ( N ∗ 0 ) 2 | x ≥ y and x +( n − 1) y ∈ n N ∗ 0 } D ( N ) = dim ϕ V ∗ ( R R ) = (1 , 1) N ∗ 0 +(0 , n ) N ∗ 0 = { ( x, y ) ∈ ( N ∗ 0 ) 2 | x ≤ y and x +( n − 1) y ∈ n N ∗ 0 } F or n = 1 , we r e c over t he situation in [5]. Note that over R al l pr oje ctive mo dules ar e dir e ct sum of inde c omp osable pr oje ctive mo dules. (ii) dim ϕ V ∗ ( R R ) = N +(0 , ∞ ) N ∗ 0 = { ( x , y ) ∈ ( N ∗ 0 ) 2 | 2 x + y ≥ 2 y + x and x +( n − 1) y ∈ n N ∗ 0 } dim ϕ V ∗ ( R R ) = D ( N )+( ∞ , 0) N ∗ 0 = { ( x, y ) ∈ ( N ∗ 0 ) 2 | 2 x + y ≤ 2 y + x and x +( n − 1) y ∈ n N ∗ 0 } In this c ase R has a sup er de c omp osable pr oje ctive right R -mo dule and a sup er de c om- p osable pr oje ctive left R -mo dule. (iii) dim ϕ V ∗ ( R R ) = N + (0 , ∞ ) N ∗ 0 = { ( x, y ) ∈ ( N ∗ 0 ) 2 | x + y ≥ 2 y and x + ( n − 1) y ∈ n N ∗ 0 } dim ϕ V ∗ ( R R ) = D ( N ) = { ( x, y ) ∈ ( N ∗ 0 ) 2 | x + y ≤ 2 y and x + ( n − 1) y ∈ n N ∗ 0 } In this situation R has a sup er de c omp osable pr oje ctive right R -mo dules but every pr o- je ctive left R -mo dule is a dir e ct sum of inde c omp osable mo dules. (iv) dim ϕ V ∗ ( R R ) = (1 , 1) N ∗ 0 + ( ∞ , 0 ) N ∗ 0 = { ( x , y ) ∈ ( N ∗ 0 ) 2 | 2 x = x + y and x ≥ y } and dim ϕ V ∗ ( R R ) = (1 , 1) N ∗ 0 = { ( x , y ) ∈ ( N ∗ 0 ) 2 | 2 x = x + y and x ≤ y } . Ther efor e, al l pr oje ctive left R -mo dules ar e fr e e henc e they ar e a dir e ct sum of finitely gener ate d mo dules but this is not tru e for pr oje ctive right R -mo dules. In p articular, V ∗ ( R R ) and V ∗ ( R R ) ar e n ot isomorphic . In the first thr e e examples V ( R ) W ( R ) = (1 , 1) N 0 + ( n, 0) N 0 ∼ = W ( R R ) . In the fourth example, as The or em 2.9 implies, V ( R ) = W ( R ) = W ( R R ) . Proof. After Theorem 1.6 wha t is left to do is to chec k the ge ne r ating sets of the monoids. But all the computations are straig h tforward. In ( iv ) to prov e that V ∗ ( R ) is not iso morphic to V ∗ ( R R ) just count the n umber o f idempo ten t elements in bo th mo noids. Remark 3.7 Examples 3.6 (ii) and (iii) answer a pr oblem mentione d in [4, p age 3261], and Example 3.6(iv) answers a pr oblem in [9, p age 310]. 16 F ol lowing the notation of Examples 3.6 and u nder the same hyp othesis, the first plac e wher e it was shown that ther e c ould b e a non finit ely gener ate d pr oje ct ive mo dule P such that dim ϕ ( h P i ) = ( n, 0) for a given n > 1 was in [20]. The monoid M = N + (0 , ∞ ) N ∗ 0 is describ e d in Examples 3.6(ii) and (iii) in two differ ent ways as a monoid given by a system of ine qualities. Both descriptions r esult in differ ent monoids D ( M ) . Now we give an example such that W ( R ) 6 ∼ = W ( R R ) and V ∗ ( R ) 6 ∼ = V ∗ ( R R ). It also shows tha t Co r ollary 2.5 fa ils a lso for the semigroup W ( R ) \ V ( R ), so that in Theorem 2.9 we cannot just assume that P is finitely gener ated mo dulo the Jaco bson radical. Example 3.8 Fix 1 ≤ n ∈ N . L et F b e any field. Ther e exist a semilo c al F-algebr a R , a suitable fi el d ext ensio n E of F and an onto ring homomorp hism ϕ : R → E × M n ( E ) su ch that Ker ϕ = J ( R ) and dim ϕ V ∗ ( R ) = (1 , n ) N ∗ 0 + · · · + (1 , 0) N ∗ 0 = { ( x , y ) ∈ ( N ∗ 0 ) 2 | nx ≥ y } dim ϕ V ∗ ( R R ) = (1 , n ) N ∗ 0 + (0 , 1) N ∗ 0 = { ( x , y ) ∈ ( N ∗ 0 ) 2 | nx ≤ y } . Ther efor e W ( R ) = (1 , n ) N 0 + · · · + (1 , 0) N 0 and W ( R R ) = (1 , n ) N 0 + (0 , 1) N 0 which ar e non isomorphi c monoids pr ovide d n ≥ 2 . Notic e that the (1 , 0) , . . . , (1 , n − 1 ) ar e minimal elements of W ( R ) and of W ( R ) \ V ( R ) so that they ar e inc omp ar able. Proof. The existence of the semilo cal ring follows from Theorem 1 .6. W e s ho w that the t wo monoids hav e the r equired set of generator s. Let M = { ( x, y ) ∈ ( N ∗ 0 ) 2 | nx ≥ y } . It is cle a r that (1 , n ) N ∗ 0 + · · · + (1 , 0) N ∗ 0 ⊆ M . If ( x, y ) ∈ M ∩ N k 0 then y = n · k + y ′ for some k , y ′ ∈ N 0 and 0 ≤ y ′ < n . Therefore , if x = k , ( x, y ) = k (1 , n ). If x > k then ( x, y ) = k (1 , n ) + ( x − k − 1 )( 1 , 0) + (1 , y ′ ) provided y ′ > 0, otherwise ( x, y ) = k (1 , n ) + ( x − k )(1 , 0). In the three ca ses we co nclude that ( x, y ) ∈ (1 , n ) N 0 + · · · + (1 , 0) N 0 . F or e lemen ts with nonempty infinite supp ort the inclusion is clear . If ( x, y ) ∈ D ( M ) ∩ N k 0 then ( x, y ) = x (1 , n ) + ( y − nx )(0 , 1) which prov es that D ( M ) = (1 , n ) N ∗ 0 + (0 , 1) N ∗ 0 . The mo noids W ( R ) and W ( R R ) have the same num b er of minimal elements if and only if n = 1. Therefo r e they canno t b e isomorphic for n ≥ 2 . 4 Monoids defined b y inequalities W e think on ( N ∗ 0 ) k and of N k 0 as order ed monoids with the order r elation given by the algebraic order. That is, ( x 1 , . . . , x k ) ≤ ( y 1 , . . . , y k ) if and only if x i ≤ y i for a n y i = 1 , . . . , k . W e recall that a monoid M is said to b e unperfor ated if, for every n ∈ N , it satisfies the following pr operties: (1) F o r a n y x , y ∈ M , nx ≤ ny implies x ≤ y ; 17 (2) for a n y x , y ∈ M , nx = ny implies x = y . where ≤ deno tes the a lgebraic pr eordering on M . Prop osition 4 . 1 ([11, Pr op osition 2]) L et A b e a c ommutative c anc el lative monoid s u ch that U ( A ) = { 0 } . Then the fol lowing st a tements ar e e quivalent; (i) A is finitely gener ate d and unp erfor ate d. (ii) Ther e exist k ≥ 1 , a monoid emb e dding f : A → N k 0 and E 1 , E 2 ∈ M ℓ × k ( N 0 ) such that f ( A ) is the set of solutions in N k 0 of the system E 1     t 1 . . . t k     = E 2     t 1 . . . t k     (iii) Ther e exist m ≥ 1 , a monoid emb e dding g : A → N m 0 , D ∈ M n × m ( N 0 ) , E 1 , E 2 ∈ M ℓ × m ( N 0 ) and m 1 , . . . , m n ∈ N , m i ≥ 2 for any i ∈ { 1 , . . . , n } , s u ch t h at g ( A ) is the set of solutions in N m 0 of t h e system D     t 1 . . . t m     ∈     m 1 N 0 . . . m n N 0     and E 1     t 1 . . . t m     = E 2     t 1 . . . t m     (iv) Ther e exist s ≥ 1 , a monoid emb e dding h : A → N s 0 , D ∈ M n × s ( N 0 ) , E 1 , E 2 ∈ M ℓ × s ( N 0 ) and m 1 , . . . , m n ∈ N , m i ≥ 2 for any i ∈ { 1 , . . . , n } , such that h ( A ) is the set of solutions in N s 0 of t h e system D     t 1 . . . t s     ∈     m 1 N 0 . . . m n N 0     and E 1     t 1 . . . t s     ≥ E 2     t 1 . . . t s     Proof. F or further quoting we give the pro of of the equiv alence of ( i i i ) a nd ( iv ). It is clear that the monoids in ( iii ) can b e descr ibed as the set of solutions of a system of congruences and inequalities a s the o nes a ppearing in ( iv ). Conv e rsely , let A b e a submonoid of N s 0 that is the set of solutions in N s 0 of the system in ( iv ). Co nsider the mono id mor phis m g : A → N s + ℓ 0 defined by g ( a 1 , . . . , a s ) = a 1 , . . . a s , s X i =1 e 1 1 i a i − s X i =1 e 2 1 i a i , . . . , s X i =1 e 1 ℓi a i − s X i =1 e 2 ℓi a i ! where ( a 1 , . . . , a s ) ∈ A and, for k = 1 , 2, e k ij denotes the i - j -ent ry of the matr ix E k . Then g ( A ) is the set o f solutions in N s + l 0 of the system D     t 1 . . . t s     ∈     m 1 N 0 . . . m n N 0     and E 1     t 1 . . . t s     = E 2     t 1 . . . t s     +     t s +1 . . . t s + ℓ     . 18 So that A is also a monoid of the type app earing in ( iii ). The embeddings of ( iii ) ar e the full a ffine embeddings. W e recall that if g ( A ) has an order unit ( n 1 , . . . , n m ) o f N m then g ( A ) can b e r e alized as dim ϕ ( V ( R )) for some semilo cal ring R such that R /J ( R ) ∼ = M n 1 ( D 1 ) × · · · × M n m ( D m ) fo r suitable divis ion rings D 1 , . . . , D m [6]. W e stress that not a ll finitely generated submonoids o f N k 0 are unp erforated. Co nsider, for example, N = (1 , 1) N 0 + (2 , 0) N 0 + (3 , 0) N 0 . In N , 2 (2 , 0) ≤ 2(3 , 0) but (2 , 0) and (3 , 0) are incompar able in N . In the next lemma we study monoids de fined by a system of equations and mono ids defined by a system of inequalities. Lemma 4.2 L et M b e a submonoid of ( N ∗ 0 ) k define d by a system of ine qu a lities D · T ∈     m 1 N ∗ 0 . . . m n N ∗ 0     ( ∗ ) and E 1 · T ≥ E 2 · T ( ∗∗ ) wher e T = ( t 1 , . . . , t k ) t , D ∈ M n × k ( N 0 ) , E 1 , E 2 ∈ M ℓ × k ( N 0 ) and m 1 , . . . , m n ∈ N , m i ≥ 2 for any i ∈ { 1 , . . . , n } . L et A b e the submonoid of M whose elements ar e the solutions in N k 0 of D · T ∈     m 1 N 0 . . . m n N 0     and E 1 · T = E 2 · T Then, (i) M and D ( M ) ar e finitely gener ate d monoids. (ii) A = M ∩ D ( M ) ∩ N k 0 . (iii) F or any m ∈ M and a ∈ A , if ther e exists m ′ ∈ ( N ∗ 0 ) k such that m = a + m ′ then m ′ ∈ M . Proof. ( i ) Consider the monoid N defined the system of equa tions D ′ · T ′ ∈     m 1 N ∗ 0 . . . m n N ∗ 0     ( ∗ ) and E 1 · T = E 2 · T +     t k +1 . . . t k + ℓ     ( ∗∗ ) where T ′ = ( t 1 , . . . , t k , t k +1 , · · · , t k + ℓ ) t and D ′ = ( D | 0) ∈ M n × ( k + ℓ ) ( N 0 ). By [10, Example 7.6], N is a finitely gener ated mono id. Let p : ( N ∗ 0 ) k + ℓ → ( N ∗ 0 ) k denote the pr o jection onto the first k co mponents. It is easy to see that p ( N ) = M , so that M is finitely ge nerated. Statement s ( ii ) and ( iii ) are clear. 19 In contrast with P ropositio n 4.1, the monoid N app earing in the pro of of Lemma 4.2 need not b e isomor phic to M . In g eneral, as the following basic example shows, a mo no id defined by inequa lities ma y not be isomor phic to a monoid defined by a system of equations. Ther efore the equiv alence of s tatemen ts (ii), (iii) and (iv) in Pr o position 4.1 does no t extend to submonoids o n ( N ∗ 0 ) k . Example 4.3 L et M b e the su bmono id of ( N ∗ 0 ) 2 that is the set of solutions of x ≥ y . Then M is not isomorphic to a monoid define d by a system of e quations. Proof. In o rder to be able to manipulate this monoid we need to think on the language of system of suppo rts, se e Definition 3.3 and Rema rk 3 .4 . First note that M = (1 , 1) N 0 + (1 , 0) N 0 + ( ∞ , 0) N 0 + ( ∞ , ∞ ) N 0 . The elements c = ( ∞ , 0) and d = ( ∞ , ∞ ) a re nonzero elements s a tisfying that 2 c = c , 2 d = d a nd d + c = d . Therefo re, if h : M → N is a monoid morphism and N is a mono id defined by a system of equations, h ( c ) and h ( d ) must be elements such that its supp ort coincides with its infinite suppo rt and, moreov er, supp h ( c ) ⊆ supp h ( d ). If h is bijective, then h ( c ) and h ( d ) are the only no n- zero elements of N such that its supp ort coincides with its infinite supp ort. So that if we think on the presentation of N as a sys tem S of supp orts, we deduce that ther e ar e only three different s ets in S , that is ∅ , supp h ( c ) and supp h ( d ). Mor eo ver, supp h ( c ) ( s upp h ( d ) On the o ther ha nd, since (1 , 0) + c = c , we deduce ∞ · h (1 , 0) = h ( c ). Similar ly , ∞ · h (1 , 1) = h ( d ). Moreov er, h (1 , 0) + h (1 , 0) 6 = h (1 , 0) and h (1 , 1) + h (1 , 0) 6 = h (1 , 1), so h (1 , 1) and h (1 , 0) hav e empt y infinite supp ort and m ust b e incompar able elements. This contradicts the fact tha t ∞ · h (1 , 1) + ∞ · h (1 , 0) = ∞ · h (1 , 1). Therefore , M cannot be isomorphic to a monoid given by equations. Finally , we draw some consequences for mono ids of pro jective mo dules of the results obtained in this section. Corollary 4.4 L et R b e a s emilo c al ring, let ϕ : R → S b e an onto ring homomorphism such that Ker ϕ = J ( R ) and S ∼ = M n 1 ( D 1 ) × · · · × M n k ( D k ) for suitable division rings D 1 , . . . , D k . A ssume that dim ϕ V ∗ ( R ) c an b e define d by a system of ine qualities such that dim ϕ V ∗ ( R R ) = D (dim ϕ V ∗ ( R )) . Then the monoids W ( R ) , W ( R R ) , V ∗ ( R ) and V ∗ ( R R ) ar e finitely gener ate d. In addi- tion, W ( R ) and W ( R R ) ar e c anc el lative and un p erfor ate d. If P is a pr oje ctive right mo dule such that h P i ∈ W ( R ) then V (End R ( P )) is a c anc el la- tive, finitely gener ate d and unp erfor ate d m o noid. Proof. By Corolla r y 2 .1 1 and Remar k 2.12. The elements o f W ( R ) a r e the so lutions in N k 0 of the system of inequalities defining M . By P ropos itio n 4.1, W ( R ) is finitely generated and unp erforated. B e ing iso morphic to a submonoid of N k 0 , W ( R ) is also cancellative. The statement on W ( R R ) follows b y symmetry . By Lemma 4.2, it follows that V ∗ ( R ) and V ∗ ( R R ) ar e finitely gener ated. Let P b e a pro jective right R -mo dule such tha t P /P J ( R ) is finitely g enerated. Using that the catego ry o f mo dules that are direct summands of P n , for some n , is equiv alent to 20 the catego r y of finitely generated pro jectiv e r igh t modules ov er End R ( P ), we deduce that V (End R ( P )) ∼ = { x ∈ W ( R ) | there exists n such that x ≤ n h P i} = M Since W ( R ) is finitely g e nerated, cancella tiv e and unp erforated then so is M . Remark 4.5 Observe that if R/J ( R ) is right no etherian t he n h P i ∈ W ( R ) if and only if P /P J ( R ) is fi nitely gener ate d. In this c ase W ( R ) is fin itely gener ate d whenever V ∗ ( R ) is finitely gener ate d. F or a g eneral semilo cal ring w e do not know whether the endo morphism ring of a pro jec- tive right R -mo dule P such that it is finitely generated modulo the Jac o bson r adical must be aga in a semilo cal ring. W e do not even know whether this ha ppens for the rings a ppear- ing in Theo rem 1.6. On the p ositive side, Corolla ry 4.4 shows tha t, at lea st, the monoid V (End R ( P )) is of the c orr e ct typ e , cf. Pr oposition 4.1. 5 Realizing monoids d efin e d by inequalities W e use the following r esult to construct semilo cal rings with prescrib ed V ∗ ( R ). Theorem 5.1 [10] L et R 1 and R 2 b e semilo c al rings, and let S = M m 1 ( D ′ 1 ) × · · · × M m ℓ ( D ′ ℓ ) for su itabl e divisio n rings D ′ 1 , . . . , D ′ ℓ . F or i = 1 , 2 , let j i : R i → S b e r ing homomorphisms. L et R b e the ring that fits into the pul lb ack diagr am R 1 j 1 − − − − → S i 1 x   x   j 2 R − − − − → i 2 R 2 Assume that j 1 is an onto ring homomorphi sm with kernel J ( R 1 ) , and that J ( R 2 ) ⊆ Ker j 2 . If R 2 /J ( R 2 ) ∼ = M n 1 ( D 1 ) × · · · × M n k ( D k ) for D 1 , . . . , D k divisio n rings, and π : R 2 → M n 1 ( D 1 ) × · · · × M n k ( D k ) is an onto morphism with kernel J ( R 2 ) then (i) i 2 induc es an onto ring homomorph ism i 2 : R → M n 1 ( D 1 ) × · · · × M n k ( D k ) with kernel J ( R ) . In p articular, R is a semilo c al ring and R /J ( R ) ∼ = R 2 /J ( R 2 ) . (ii) L et α : dim π V ∗ ( R 2 ) → ( N ∗ 0 ) ℓ b e the monoid homomorphism induc e d by j 2 . Then dim i 2 V ∗ ( R ) = { x ∈ dim π V ∗ ( R 2 ) | α ( x ) ∈ dim j 1 V ∗ ( R 1 ) } , and dim i 2 V ∗ ( R R ) = { x ∈ dim π V ∗ ( R 2 R 2 ) | α ( x ) ∈ dim j 1 V ∗ ( R 1 R 1 ) } . Example 5.2 L et k ∈ N , and let a 1 , . . . , a k , b 1 , . . . , b k ∈ N 0 . L et ( n 1 , . . . , n k ) ∈ N k b e such that a 1 n 1 + · · · + a k n k = b 1 n 1 + · · · + b k n k ∈ N . F or any field extension F ⊆ F 1 , ther e exist a semilo c al F - al gebr a R and an onto morphism of F - a lgebr as ϕ : R → M n 1 ( F 1 ) × · · · × M n k ( F 1 ) 21 with kernel J ( R ) such that dim ϕ V ∗ ( R R ) is the set of solutions in ( N ∗ 0 ) k of the ine quality a 1 t 1 + · · · + a k t k ≥ b 1 t 1 + · · · + b k t k and dim ϕ V ∗ ( R R ) is t he set of solutions in ( N ∗ 0 ) k of the ine quality a 1 t 1 + · · · + a k t k ≤ b 1 t 1 + · · · + b k t k . Note that dim ϕ ( h R i ) = ( n 1 , . . . , n k ) . Proof. Se t m = a 1 n 1 + · · · + a k n k = b 1 n 1 + · · · + b k n k . Let T b e a semilo cal F -alg e bra with an ont o algebra morphis m j 1 : T → F 1 × F 1 with Ker( j 1 ) = J ( T ), and suc h that dim j 1 V ∗ ( T T ) = { ( x, y ) ∈ ( N ∗ 0 ) 2 | x ≥ y } and dim j 1 V ∗ ( T T ) = { ( x, y ) ∈ ( N ∗ 0 ) 2 | y ≥ x } . Such T exists by The o rem 3.1. Let M m ( j 1 ) : M m ( T ) → M m ( F 1 ) × M m ( F 1 ) b e the induced morphism. Set R 2 = M n 1 ( F 1 ) × · · · × M n k ( F 1 ). Consider the morphism of F -a lgebras j 2 : R 2 − → M m ( F 1 ) × M m ( F 1 ) defined by j 2 ( r 1 , . . . , r k ) =                   r 1 ··· 0 . . . . . . a 1 ) . . . 0 ·· · r 1 · · · 0 . . . 0 · · · r k ··· 0 . . . . . . a k ) . . . 0 ··· r k          ,          r 1 ··· 0 . . . . . . b 1 ) . . . 0 ··· r 1 · · · 0 . . . 0 · · · r k ··· 0 . . . . . . b k ) . . . 0 ·· · r k                   The morphism j 2 induces the morphism of monoids f : ( N ∗ 0 ) k → N ∗ 0 × N ∗ 0 defined by f ( x 1 , . . . , x k ) = ( a 1 x 1 + · · · + a k x k , b 1 x 1 + · · · + b k x k ). Hence, f ( n 1 , . . . , n k ) = ( m, m ). Let R b e the ring defined b y the pullback diagram M m ( T ) M m ( j 1 ) − − − − − → M m ( F 1 ) × M m ( F 1 ) i 1 x   x   j 2 R − − − − → ϕ M n 1 ( F 1 ) × · · · × M n k ( F 1 ) Applying Theorem 5 .1 (i), we conclude that R is a s e milocal F -a lgebra and that ϕ is an onto mor phism of F -algebra s with kernel J ( R ). By Theorem 5.1 (ii), ( x 1 , . . . , x k ) ∈ dim ϕ V ∗ ( R R ) if and o nly if f ( x 1 , . . . , x k ) ∈ dim M m ( j 1 ) V ∗ ( M m ( T )) if a nd only if a 1 x 1 + · · · + a k x k ≥ b 1 x 1 + · · · + b k x k . Similarly , ( x 1 , . . . , x k ) ∈ dim ϕ V ∗ ( R R ) if a nd only if a 1 x 1 + · · · + a k x k ≤ b 1 x 1 + · · · + b k x k . Now we are ready to prove Theorem 1.6. Proof of Theorem 1.6. Let M b e the monoid defined b y the sys tem of inequalities, D     t 1 . . . t k     ∈     m 1 N ∗ 0 . . . m n N ∗ 0     ( ∗ ) and E 1     t 1 . . . t k     ≤ E 2     t 1 . . . t k     ( ∗∗ ) where D ∈ M n × k ( N 0 ), E 1 , E 2 ∈ M ℓ × k ( N 0 ), n, ℓ ≥ 0 a nd m 1 , . . . , m n ∈ N , m i ≥ 2 for any i ∈ { 1 , . . . , n } . By [10, Theore m 5.3] we know the following. 22 Step 1. Ther e exist a field extension E of F , a (no etherian) semilo c al F -algebr a R 1 and an onto morphism of F - a lgebr as ϕ 1 : R 1 → M n 1 ( E ) × · · · × M n k ( E ) s u ch t h at dim ϕ 1 V ∗ ( R 1 ) is the set of solutions in ( N ∗ 0 ) k of the system of c ongruen c es ( ∗ ) . Now we need to prov e, Step 2. Ther e exist a semilo c al F -algebr a R 2 and an onto morphism of F -algebr as ϕ 2 : R 2 → M n 1 ( E ) × · · · × M n k ( E ) such that dim ϕ 2 V ∗ ( R 2 ) is the set of solut io ns in ( N ∗ 0 ) k of the system of ine qualities ( ∗∗ ) and dim ϕ 2 V ∗ ( R 2 R 2 ) is the set of solutions in ( N ∗ 0 ) k of the syst em of ine qualities D ( ∗∗ ) . If ℓ = 0, that is, if ( ∗∗ ) is empty we set R 2 = M n 1 ( E ) × · · · × M n k ( E ) and ϕ 2 = Id. Assume ℓ > 0. Therefo re, we can assume that no ne of the rows in E 1 and, hence, in E 2 are zero. By Example 5 .2, for i = 1 , . . . , ℓ , there exist a no e th erian s emilocal F -a lgebra T i and an ont o mor phism o f F -algebr as π i : T i → M n 1 ( E ) × · · · × M n k ( E ) with kernel J ( T i ) a nd such that dim π i V ∗ ( T i ) is the set of solutions in ( N ∗ 0 ) k of the i -th inequality defined by the matrices E 1 and E 2 , and dim π i V ∗ ( T i T i ) is the set of so lutions in ( N ∗ 0 ) k of the reversed inequality . Let R 2 be the pullback of π i , i = 1 , . . . , ℓ . By Theorem 5.1, R 2 is a semilo cal F -algebr a with an onto morphism of F -algebr as ϕ 2 : R 2 → M n 1 ( E ) × · · · × M n k ( E ) with kernel J ( R 2 ). Moreov er, dim ϕ 2 V ∗ ( R 2 ) is the set of solutions of the inequalities ( ∗∗ ) and dim ϕ 2 V ∗ ( R 2 R 2 ) is the set of solutions of the inequa lities D ( ∗∗ ). This concludes the pr oof of Step 2. Finally , se t R to b e the pullback of ϕ i : R i → M n 1 ( E ) × · · · × M n k ( E ), i = 1 , 2. 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