Spectral Homotopy and the Spectral Fundamental Group

SPECTRAL HOMOTOPY AND THE SPECTRAL FUND AMENT AL GR OUP BISW AJIT MITRA AND SOURA V K ONER Abstract. In this pap er, we introduce an algebraic-top ological inv arian t for commutativ e pm-rings, termed the spectral fundamental group, whic h is denoted b y π alg k ( A ). This group is defined via homotop y classes of loops within the space of induced sp ectral maps, which are generated b y the k - algebra endomorphism monoid of the ring. W e establish foundational prop- erties of this inv ariant, pro ving that π alg k ( A ) is an ab elian group that natu- rally resp ects direct products and admits natural morphisms with resp ect to fully inv ariant subrings. F urther, we establish an explicit isomorphism between the sp ectral fundamental group of certain contin uous function rings and the classical fundamen tal group of their associated topological mapping spaces. Finally , utilizing a generalized dual num b er construction, we presen t an explicit example of a pm-ring that cannot b e embedded in to any function ring o ver a field of characteristic zero, yet possesses a nontriv- ial sp ectral fundamental group. This demonstrates that π alg k ( A ) captures homotopical dynamics that are in trinsically algebraic. 1. Introduction F or a comm utativ e ring A with unit y , the prime sp ectrum Sp ec( A ) and the maximal spectrum Max( A ) pro vide a fundamen tal bridge b et w een algebra and top ology . In particular, for pm-rings—rings in which ev ery prime ideal is con- tained in a unique maximal ideal—the canonical retraction µ : Spec( A ) → Max( A ) defined by assigning to eac h prime ideal the unique maximal ideal containing it, induces a natural and w ell-b eha v ed top ological structure on M ax ( A ) This pro vides a conv enient framew ork for studying the interpla y betw een algebraic prop erties of the ring and the associated topological structure of its prime and maximal sp ectra. 2020 Mathematics Subje ct Classific ation. Primary 54H13, 13J99; Secondary 55Q05, 13A15, 54C35. Key wor ds and phr ases. sp ectral fundamen tal group; pm-rings; spectral homotop y; induced spectral maps; maximal sp ectra; rings of con tinuous functions. 1 2 BISW AJIT MITRA AND SOURA V K ONER A cen tral theme in b oth comm utative algebra and topology is the study of in v ariants that enco de structural information. The goal of this pap er is to in- tro duce and study suc h an inv arian t. Sp ecifically , w e consider the space of self-maps of Max( A ) that are induced by k -algebra endomorphisms of A , and w e inv estigate its homotopical structure. This leads to the definition of a new in v ariant, whic h we call the sp ectral fundamental group, denoted by π alg k ( A ). By construction, this group enco des homotopy classes of lo ops in the space of induced sp ectral maps, thereby measuring the existence of nontrivial con tin uous one-parameter families of algebra endomorphisms. This persp ectiv e places our work at the intersection of commutativ e algebra and homotopy theory . The construction is closely related to classical top ology: in the case of rings of contin uous functions, we show that π alg k ( A ) recov ers the fundamen tal group of a mapping space. More precisely , for a compact Hausdorff space X , we obtain a natural isomorphism π alg k ( C ( X )) ∼ = π 1 ( C ( X, X ) , id X ) , thereb y situating our inv ariant within the well-studied top ology of function spaces. Despite this connection, the sp ectral fundamental group is not merely a reform ulation of known topological inv arian ts. A key feature of our construction is that it is defined purely in terms of algebraic endomorphisms, and therefore re- mains meaningful even for rings that do not arise from classical function spaces. T o demonstrate this, we construct explicit examples of pm-rings that cannot b e em b edded into any function ring ov er a field of c haracteristic zero, yet admit a non trivial sp ectral fundamental group. These examples show that π alg k ( A ) can detect gen uinely algebraic phenomena that are not accessible through traditional top ological mo dels. The main contributions of this pap er are as follows: W e define the sp ectral fundamen tal group π alg k ( A ) using homotopy classes of lo ops in the space of induced spectral maps. W e pro ve that this in v arian t carries a natural abelian group structure and b ehav es w ell with respect to standard constructions, including direct pro ducts and fully inv arian t retracts. W e estab- lish a precise connection with classical top ology by identifying π alg k ( C ( X )) with the fundamental group of the mapping space C ( X, X ). W e construct explicit examples of pm-rings exhibiting nontrivial sp ectral fundamental groups b ey ond the realm of function rings, thereby demonstrating the in trinsically algebraic nature of the inv ariant. The paper is organized as follows. In Section 2, we review the necessary bac kground on pm-rings and top ological function spaces. In Section 3, we in- tro duce endomorphism monoids, induced sp ectral maps, and the definition of the sp ectral fundamen tal group. Section 4 develops the general structure the- ory of this inv ariant, including pro duct b ehavior. Finally , Section 5 is devoted to constructions that illustrate the scop e of the theory b ey ond classical function SPECTRAL FUNDAMENT AL GROUP 3 spaces. Through this w ork, we aim to initiate a systematic study of homotopical in v ariants arising from algebraic endomorphisms, and to highlight a new av en ue for interaction b et w een commutativ e algebra and top ology . 2. Preliminaries F or a commutativ e ring A with unit y , let Sp ec( A ) and Max( A ) denote the prime and maximal sp ectra of A , resp ectiv ely . Both Sp ec( A ) and Max( A ) are endo wed with the Zariski top ology . A comm utative ring with unity is said to b e a pm-ring if every prime ideal is contained in a unique maximal ideal. The result b elow is well-kno wn and can b e found, for instance, in [3] and [4]. Theorem 2.1. The fol lowing ar e e quivalent. ( a ) A is a pm-ring. ( b ) Max( A ) is a r etr act of Sp ec( A ) . ( c ) Whenever a + b = 1 in A , ther e exist r, s ∈ A such that (1 − ar )(1 − bs ) = 0 . F or a pm-ring A , it follows from part ( c ) of Theorem (2.1) that Max( A ) is a Hausdorff space. Moreo v er, the map (2.1) µ : Spec( A ) − → Max( A ) , whic h assigns to each prime ideal P the unique maximal ideal µ ( P ) containing P , is the unique retraction of Sp ec( A ) onto Max( A ). In what follows, w e consider spaces of con tin uous maps in v olving pro ducts of topological spaces and their asso ciated function spaces. In particular, we make use of a fundamental result describing the relationship b etw een contin uous maps on pro duct spaces and con tinuous maps into function spaces. F or the reader’s conv enience, we recall this result b elo w, which can b e found, for example, in [2]. Theorem 2.2 (Exp onential Law) . L et X b e a lo c al ly c omp act Hausdorff sp ac e and let Y and Z b e top olo gic al sp ac es. Then C ( X × Z, Y ) ∼ = C  Z, C ( X , Y )  , wher e al l function sp ac es ar e endowe d with the c omp act-op en top olo gy. 3. Endomorphisms and Induced Spectral Maps Throughout this pap er, we assume k is one of the following rings: Z or R . Let k - PM denote the category whose ob jects are pm-rings equipp ed with a k - algebra structure, and whose morphisms are unital k -algebra homomorphisms. F or A ∈ Ob( k - PM ), we write E k ( A ) := Hom k - PM ( A, A ) for its endomorphism monoid, that is, under comp osition, E k ( A ) is a monoid with identit y id A . F or f ∈ E k ( A ), the map f ∗ : Sp ec( A ) → Sp ec( A ), defined by P 7→ f − 1 ( P ), is contin uous. Consequently , the restriction of f ∗ to Max( A ) is 4 BISW AJIT MITRA AND SOURA V K ONER also con tinuous. By (2.1), since µ is the unique retraction, it follo ws immediately that µ f := µ ◦ f ∗   Max( A ) is contin uous. Adopting this terminology , w e make the follo wing definition. Definition 3.1 (Induced Sp ectral Maps) . F or A ∈ Ob( k - PM ), define X A k := { µ f | f ∈ E k ( A ) } and Y := C (Max( A ) , Max( A )) . The elemen ts of X A k are called induced sp ectral maps. W e endow the space Y with the compact-op en top ology and X A k with the induced subspace top ology . 3.1. Homotop y of Sp ectral Lo ops. Definition 3.2 (Homotop y of Induced Sp ectral Maps) . Two induced spectral maps µ f and µ g in X A k are said to b e homotopic, written µ f ≃ µ g , if there exists a contin uous path α : [0 , 1] → X A k suc h that α (0) = µ f and α (1) = µ g . By Theorem (2.2), this amoun ts to stating that there exists a con tinuous map e α : Max( A ) × [0 , 1] → Max( A ) such that e α ( − , t ) = µ h t for some h t ∈ E k ( A ) with h 0 = f and h 1 = g . In this case, we also say that f and g are sp ectrally homotopic. Definition 3.3 (Spectral Lo op at µ id A ) . A sp ectral lo op at µ id A is a con tin uous map γ : [0 , 1] → X A k suc h that γ (0) = γ (1) = µ id A . Definition 3.4 (Homotopy of Sp ectral Loops at µ id A ) . Two sp ectral lo ops γ and η are said to b e homotopic if there exists a contin uous map F : [0 , 1] × [0 , 1] → X A k suc h that (1) F ( − , 0) = γ and F ( − , 1) = η , (2) F (0 , s ) = F (1 , s ) = µ id A for all s ∈ [0 , 1]. 3.2. Sp ectral F undamental Group. Definition 3.5. The sp ectral fundamental group of A , written π alg k ( A ), is the fundamen tal group π 1 ( X A k , µ id A ), that is, π alg k ( A ) := { sp ectral lo ops at µ id A }  homotop y . The group op eration is induced by concatenation of lo ops, and the identit y elemen t is the constant lo op at µ id A . SPECTRAL FUNDAMENT AL GROUP 5 4. General Resul ts Theorem 4.1. π alg k ( A ) is an ab elian gr oup. Pr o of. The space X A k is a submonoid of Y under comp osition. Since Max( A ) is a compact Hausdorff space, composition is contin uous in the compact-op en top ology , so X A k is a topological monoid. Thus, π alg k ( A ) is the fundamental group of this H-space based at its identit y elemen t µ id A . By the Eckmann–Hilton argumen t, the fundamental group of any H-space at its identit y is ab elian. ■ Recall that a k -subalgebra D of B is said to b e fully in v arian t in B if f ( D ) ⊆ D for ev ery f ∈ E k ( B ). Also, D is said to b e a retract of B if there exists a morphism r : B → D such that r | D is the identit y map on D . Theorem 4.2. L et h : A → B b e a morphism with the kernel I , and supp ose that h ( A ) is ful ly invariant in B . Then ther e exists a natur al morphism h ∗ fr om π alg k ( B ) to π alg k ( A/I ) . Mor e over, if h is an isomorphism, then so is h ∗ . Pr o of. Let h ( A ) = D . Since A/I ∼ = D , let g : A/I → D b e the isomorphism and let θ : Max( D ) → Max( A/I ) b e the homeomorphism, induced by h and g resp ectiv ely . Let δ : Max( B ) → Max( D ) b e a map which sends every maximal ideal M of B into the unique maximal ideal of D containing M ∩ D . By theorem (1 . 6) in [3], δ is a contin uous surjection. If σ : Max( B ) → Max( A/I ) is the map suc h that σ = ( θ ◦ δ ), then it is easy to see that σ is also a contin uous surjection. Supp ose now that f ∈ E k ( B ) and M 0 ∈ Max( B ). Let µ f ( M 0 ) = M 1 , δ ( M 1 ) = M 2 , and θ ( M 2 ) = M 3 . Also, let δ ( M 0 ) = M 4 , θ ( M 4 ) = M 5 , and µ g − 1 ◦ f | D ◦ g ( M 5 ) = M 6 . This implies the following (1) f − 1 ( M 0 ) ⊆ M 1 , M 1 ∩ D ⊆ M 2 , and g − 1 ( M 2 ) = M 3 ; (2) M 0 ∩ D ⊆ M 4 and g − 1 ( f − 1 ( M 4 )) ⊆ M 6 . F rom (1), we conclude that g − 1 ( f − 1 ( M 0 )) ∩ A/I ⊆ g − 1 ( M 1 ) ∩ A/I ⊆ g − 1 ( M 2 ) ⊆ M 3 . As D is fully inv ariant in B , therefore from (2), we obtain that g − 1 ( f − 1 ( M 0 )) ∩ A/I ⊆ g − 1 ( f − 1 ( M 4 )) ⊆ M 6 . Since the homomorphic image of a pm-ring is a pm-ring, therefore, we conclude that M 3 = M 6 , that is, for each f ∈ E k ( B ), the rectangular diagram in (4.1) comm utes. (4.1) Max( B ) Max( B ) Max( A/I ) Max( A/I ) µ f σ σ µ ( g − 1 ◦ f | D ◦ g ) 6 BISW AJIT MITRA AND SOURA V K ONER Define a map ψ : X B k → X A/I k b y µ f 7→ µ g − 1 ◦ f | D ◦ g . Let [ K, U ] b e a subbasic op en set containing µ g − 1 ◦ f | D ◦ g . Consider the subbasic op en set [ σ − 1 ( K ) , σ − 1 ( U )]. By the rectangular diagram in (4.1), we conclude that µ f ∈ [ σ − 1 ( K ) , σ − 1 ( U )], and that µ g − 1 ◦ l | D ◦ g ∈ [ K , U ] for ev ery µ l ∈ [ σ − 1 ( K ) , σ − 1 ( U )]. This shows that ψ is con tinuous. Finally , if w e define the map h ∗ : π alg k ( B ) → π alg k ( A/I ) giv en b y [ µ f ] 7→ [ µ g − 1 ◦ f | D ◦ g ], then the contin uit y of ψ pro ves that h ∗ is a well-defined map. Moreo ver, it is evident that h ∗ is a group homomorphism. If h is an isomorphism, then b y considering h − 1 : B → A and b y applying the first part, we get a group homomorphism ( h − 1 ) ∗ : π alg k ( A ) → π alg k ( B ), which serv es as the inv erse of h ∗ . This completes the pro of. ■ Theorem 4.3. If D is a ful ly invariant r etr act of B , then the natur al morphism i ∗ fr om π alg k ( B ) to π alg k ( D ) is surje ctive. Mor e over, π alg k ( B ) ∼ = π alg k ( D ) × k er( i ∗ ) . Pr o of. By Theorem (4.2), there is a natural morphism i ∗ : π alg k ( B ) → π alg k ( D ) giv e by [ µ f ] 7→ [ µ f | D ]. Supp ose now that [ µ l ] ∈ π alg k ( D ). Let r : B → D b e a retraction. Since [ µ l ◦ r ] ∈ π alg k ( B ) and since ( l ◦ r ) | D = l , we conclude that i ∗ is a surjective morphism. Supp ose that f ∈ E k ( D ). Let r ∗ : Sp ec( D ) → Sp ec( B ) be the contin uous map induced by the retraction r and let θ r : Max( D ) → Max( B ) b e the map given b y θ r = µ ◦ r ∗ | Max( D ) , where µ : Sp ec( B ) → Max( B ) is the unique retraction [3]. Also, let i : D  → B be the inclusion. Supp ose M 1 ∈ Max( D ) b e any . Let µ f ( M 1 ) = M 2 , θ r ( M 2 ) = M 3 , θ r ( M 1 ) = M 4 , and µ ( i ◦ f ◦ r ) ( M 4 ) = M 5 . This implies the following: (1) f − 1 ( M 1 ) ⊆ M 2 and r − 1 ( M 2 ) ⊆ M 3 ; (2) r − 1 ( M 1 ) ⊆ M 4 and r − 1 ( f − 1 ( i − 1 ( M 4 ))) ⊆ M 5 . F rom (1), we conclude that r − 1 ( f − 1 ( M 1 )) ⊆ r − 1 ( M 2 ) ⊆ M 3 . F rom (2), we conclude that r − 1 ( f − 1 ( i − 1 ( r − 1 ( M 1 )))) ⊆ r − 1 ( f − 1 ( i − 1 ( M 4 ))) ⊆ M 5 . Notice no w that f ◦ r = r ◦ i ◦ f ◦ r . Thus, w e get that M 3 = M 5 , that is, for eac h f ∈ E k ( D ), the rectangular diagram in (4.2) commutes. (4.2) Max( D ) Max( D ) Max( B ) Max( B ) µ f θ r θ r µ ( i ◦ f ◦ r ) SPECTRAL FUNDAMENT AL GROUP 7 Define a map ψ : X D k → X B k b y µ f 7→ µ ( i ◦ f ◦ r ) . If [ K, U ] be a subbasic op en set containing µ ( i ◦ f ◦ r ) , then it is easy to see that [ θ − 1 r ( K ) , θ − 1 r ( U )] is a subbasic op en set con taining µ f . Moreo ver, for eac h µ g ∈ [ θ − 1 r ( K ) , θ − 1 r ( U )], w e ha ve that µ ( i ◦ g ◦ r ) ∈ [ K, U ]. This shows that the map ψ is contin uous. Hence, the map t : π alg k ( D ) → π alg k ( B ) defined by [ µ f ] 7→ [ µ ( i ◦ f ◦ r ) ] is a well-defin ed group homomorphism. If 1 is the iden tit y map on π alg k ( D ), then observ e that i ∗ ◦ t = 1 , that is, t is a lift to the iden tity map 1 . Thus, π alg k ( B ) ∼ = π alg k ( D ) × ker( i ∗ ). ■ The following result follo ws trivially from Theorem (4.2) and Theorem (4.3), so the pro of is omitted. Theorem 4.4. L et h : A → B b e a morphism with the kernel I , and supp ose that h ( A ) is a ful ly invariant r etr act of B . Then the natur al morphism h ∗ fr om π alg k ( B ) to π alg k ( A/I ) is surje ctive. Mor e over, π alg k ( B ) ∼ = π alg k ( A/I ) × k er( h ∗ ) . A central goal in comm utative ring theory is understanding how inv arian ts b eha ve under standard ring constructions. The group π alg k naturally respects the direct pro duct. This structural corresp ondence is formalized in the following. Theorem 4.5. π alg k ( A × B ) ∼ = π alg k ( A ) × π alg k ( B ) . Pr o of. Let Π 1 : A × B → A and Π 2 : A × B → B be the pro jections. Observ e first that for eac h f ∈ E k ( A × B ), Π 1 ◦ f = f 1 ∈ E k ( A ) and Π 2 ◦ f = f 2 ∈ E k ( B ). Con versely , if h 1 ∈ E k ( A ) and h 2 ∈ E k ( B ), then h = ( h 1 , h 2 ) : A × B → A × B defined by ( a, b ) 7→ ( h 1 ( a ) , h 2 ( b )) is a morphism. Thus, there exists a one-to-one corresp ondence b et w een the sets E k ( A × B ) and E k ( A ) × E k ( B ). Let f ∈ E k ( A × B ), Π 1 ( f ) = f 1 , and Π 2 ( f ) = f 2 . As mentioned earlier, the map µ f : Max( A × B ) → Max( A × B ), given by M × B 7→ µ f 1 ( M ) × B and A × N 7→ A × µ f 2 ( N ), is contin uous. Let σ : Max( A × B ) → Max( A ) ⊔ Max( B ) be the homeomorphism giv en b y M × B 7→ M and A × N 7→ N . It is straightforw ard to see that the map µ f 1 F µ f 2 : Max( A ) ⊔ Max( B ) → Max( A ) ⊔ Max( B ), giv en b y M 7→ µ f 1 ( M ) and N 7→ µ f 2 ( N ), is con tinuous. Since σ ◦ µ f = ( µ f 1 F µ f 2 ) ◦ σ , therefore, w e conclude that for eac h f ∈ E k ( A × B ), the rectangular diagram in (4.3) commutes. (4.3) Max( A × B ) Max( A × B ) Max( A ) ⊔ Max( B ) Max( A ) ⊔ Max( B ) µ f σ σ µ f 1 F µ f 2 Define a map ψ : X A × B k → X A k × X B k b y ψ ( µ f ) = ( µ f 1 , µ f 2 ). Since the sets E k ( A × B ) and E k ( A ) × E k ( B ) are in a one-to-one c orrespondence, we conclude 8 BISW AJIT MITRA AND SOURA V K ONER that ψ is bijective. Finally , if we consider [ K A , U A ] × [ K B , U B ], the subbasic op en set containing ( µ f 1 , µ f 2 ), then by the diagram in (4.3), w e conclude that µ f ∈ [ σ − 1 ( K A ∪ K B ) , σ − 1 ( U A ∪ U B )]. This simultaneously shows that ψ is b oth a contin uous and an op en map with ψ ( µ id A × B ) = ( µ id A , µ id B ). ■ 4.1. T rivialit y of π alg k ( A ) . If A is a clean ring and if γ : [0 , 1] → X A k is a sp ectral lo op, then γ is constant, since Max( A ) is totally disconnected, and so π alg k ( A ) = 0. Thus, for a top ological space X , if A is either of the rings C c ( X ), C ∗ c ( X ) or C F ( X ), then we can conclude that π alg k ( A ) = 0. F or a given ring A , the group π alg k ( A ) consists of homotopy classes of lo ops in the space of self-maps of Max( A ) that arise from morphisms of A . Since every in termediate stage of a spectral homotopy must remain in X A k , suc h lo ops are constrained b y the rigidity of morphisms. So, π alg k ( A ) measures the exten t to whic h A admits contin uous one-parameter families of morphisms. F or sp ectrally rigid rings, the sp ectral fundamental group is trivial. It b ecomes nontrivial pre- cisely when morphisms produce non-n ull-homotopic lo ops in the space of induced sp ectral maps. In this wa y , π alg k ( A ) records homotopical dynamics intrinsic to the algebraic structure of A . 4.2. Non trivialit y of π alg k ( A ) . Theorem 4.6. L et X b e a top olo gic al sp ac e. Then for some T ychonoff sp ac e Y , π alg R  C ∗ ( X )  ∼ = π alg Z  C ∗ ( X )  ∼ = π 1  C ( β Y , β Y ) , id β Y  . Pr o of. By theorem (3 . 9) in [1], there exists a Tyc honoff space Y suc h that C ( X ) ∼ = C ( Y ); moreov er, this isomorphism maps C ∗ ( X ) onto C ∗ ( Y ). Th us, applying Theorem (4.2), w e get that π alg Z ( C ∗ ( X )) ∼ = π alg Z ( C ∗ ( Y )). F urther, since C ∗ ( Y ) ∼ = C ( β Y ), therefore again applying Theorem (4.2), we get that π alg Z ( C ∗ ( Y )) ∼ = π alg Z ( C ( β Y )). Let A = C ( β Y ). Since Max( A ) ∼ = β Y , let θ : Max( A ) → β Y b e the homeomorphism given b y M p 7→ p . By theorem (10 . 6) in [1], eac h f ∈ E ( A ) is induced by a unique con tinuous map φ f : β Y → β Y . If µ f : Max( A ) → Max( A ) is the contin uous map induced b y f ∈ E Z ( A ), then µ f ( M p ) = M ϕ f ( p ) . This amounts to stating that the rectangular diagram in (4.4) commutes. (4.4) Max( A ) Max( A ) β Y β Y µ f θ θ ϕ f SPECTRAL FUNDAMENT AL GROUP 9 W e now sho w that X A Z ∼ = C ( β Y , β Y ). Consider the map χ : X A Z → C ( β Y , β Y ) giv en by χ ( µ f ) = φ f for all f ∈ E Z ( A ). It is evident that χ is bijective. Now, for f ∈ E Z ( A ), supp ose that φ f ∈ [ K, U ] where K is compact a nd U is op en in β Y . If we consider the subbasic op en set [ θ − 1 ( K ) , θ − 1 ( U )], then by the diagram in (4.4), µ f ∈ [ θ − 1 ( K ) , θ − 1 ( U )] and χ ([ θ − 1 ( K ) , θ − 1 ( U )]) = [ K, U ]. This simultaneously shows that χ is b oth a contin uous and an op en map with χ ( µ id A ) = id β Y . Finally , since every unital ring homomorphism from C ( X ) to itself is an R - algebra homomorphism, we conclude that π alg R  C ∗ ( X )  ∼ = π alg Z  C ∗ ( X )  . ■ Corollary 4.7. If X is a compact Hausdorff space, then π alg R  C ( X )  ∼ = π alg Z  C ( X )  ∼ = π 1  C ( X, X ) , id X  . Pr o of. In this case, C ( X ) ∼ = C ∗ ( X ) and β X ∼ = X . Th us, applying Theorem (4.6), the result follows. ■ Example 4.8. Let A = C ( S 1 ), where S 1 denotes the unit circle. By Corollary (4.7), we hav e π alg Z ( A ) ∼ = π 1 ( C ( S 1 , S 1 ) , id S 1 ) . It is well-kno wn that π 1 ( C ( S 1 , S 1 ) , id S 1 ) ∼ = Z . Hence, π alg k ( C ( S 1 )) ∼ = Z . 5. Spectral fundament al groups beyond function sp aces Throughout this section, we w ork with the category R - PM . Let T b e a path- connected compact Hausdorff space. Motiv ated by Corollary (4.7), it is natural to ask the following questions: First, do es there exist a pm-ring A that cannot b e realized as a subring of a function ring ov er a field of characteristic zero and suc h that Max( A ) ∼ = T ? Second, do es there exist a pm-ring A that cannot b e realized as a subring of a function ring ov er a field of characteristic zero, y et p ossesses a nontrivial sp ectral fundamental group? One might initially exp ect that such rings can only b e realized as subrings of a function ring ov er a field of characteristic zero. How ever, our first theorem utilizes a generalized dual n um b er construction to provide an explicit example of a pm-ring whose maximal sp ectrum is homeomorphic to a given top ological space, but which fails to embed algebraically in to an y function ring ov er a field of c haracteristic zero. F urthermore, our second theorem provides an explicit example of an algebraic pm-ring with a nontrivial sp ectral fundamental group. Let C ( T ) be the ring of all real-v alued contin uous functions on T and let B = C ( T ). Let ∆ be an y set. F or eac h δ ∈ ∆ let n δ ∈ N be an y suc h that n δ > 1. Throughout this section, w e w ork with the ring A = B [ x δ : δ ∈ ∆] / ( x n δ δ : δ ∈ ∆). Theorem 5.1. A c annot b e emb e dde d into any function ring over a field of char acteristic zer o, yet Max( A ) is a p ath-c onne cte d c omp act Hausdorff sp ac e with | Max( A ) | > 1 . 10 BISW AJIT MITRA AND SOURA V K ONER Pr o of. Since A is not reduced, so it is not isomorphic to an y subring of a function ring ov er a field of characteristic zero. Let us consider the ideal I = ( x δ : δ ∈ ∆) / ( x n δ δ : δ ∈ ∆) of A . Since A/I ∼ = B , there is a surjective ring homomorphism ϕ : A → B with k ernel I . Observe that N( A ) = I . As I ⊆ J( A ), we conclude that ϕ − 1 (J( B )) = J( A ). Also, since J( B ) = 0, we obtain that J( A ) = I . The map ϕ ∗ : Max( B ) → Max( A ) is a homeomorphism is well-kno wn, since ϕ is surjective and V ( I ) = Max( A ). No w, if we take T as a path-connected compact Hausdorff space with | T | > 1, then Max( A ) ∼ = T ∼ = Max( B ). F urther, since Max( A ) is Hausdorff and since J( A ) = N( A ), we conclude that A is a pm-ring (see [4]). ■ Theorem 5.2. If π alg R ( B )  = 0 , then π alg R ( A )  = 0 . Pr o of. There exists a unique morphism ˜ r : B [ x δ : δ ∈ ∆] → B that extends the iden tity map on B and satisfies ˜ r ( x δ ) = 0 for all δ ∈ ∆. Let J = ( x n δ δ : δ ∈ ∆). F or any δ ∈ ∆, we hav e ˜ r ( x n δ δ ) = ( ˜ r ( x δ )) n δ = 0. Thus, ˜ r ( J ) = { 0 } . This shows that ˜ r factors through the canonical pro jection π : B [ x δ : δ ∈ ∆] → A , inducing a w ell-defined morphism r : A → B such that r ◦ π = ˜ r . F or an y b ∈ B , viewed as an element of A , we hav e r ( b ) = ˜ r ( b ) = b . Therefore, r | B is the identit y map on B , proving that B is a retract of A . Let σ : A → A b e an arbitrary morphism. Then A p ossesses a natural grading b y total degree, that is, A = ∞ M j =0 A j , where A 0 = B , and for j ≥ 1, A j is the free B -mo dule generated by the set M j =  Y ¯ x α δ δ : X α δ = j, 0 ≤ α δ < n δ  . Assume, for the sake of con tradiction, that σ ( B ) ⊆ B . This means there exists some b ∈ B such that σ ( b ) has a non-zero comp onent in A j for some j ≥ 1. Let k ≥ 1 be the minimal integer such that there exists some b ∈ B where the degree k comp onent of σ ( b ) is non-zero. By the minimality of k , for an y c ∈ B , the pro jection of σ ( c ) onto A j is identically zero for all 0 < j < k . Thus, for any c ∈ B , w e can write σ ( c ) = σ 0 ( c ) + σ k ( c ) + R ( c ), where σ 0 ( c ) ∈ B , σ k ( c ) ∈ A k , and R ( c ) ∈ L j >k A j . Since σ is a morphism, for an y b, c ∈ B , we ha ve σ ( bc ) = σ ( b ) σ ( c ). Expanding b oth sides yields the following: σ 0 ( bc ) + σ k ( bc ) + R ( bc ) =  σ 0 ( b ) + σ k ( b ) + R ( b )  σ 0 ( c ) + σ k ( c ) + R ( c )  . Multiplying the right side and collecting terms by degree, we s ee the degree zero term is σ 0 ( b ) σ 0 ( c ). As σ ( b + c ) = σ ( b ) + σ ( c ), therefore we obtain that σ 0 ( b + c ) = σ 0 ( b ) + σ 0 ( c ). Hence, σ 0 : B → B is a morphism. F urther, the SPECTRAL FUNDAMENT AL GROUP 11 degree k term is σ 0 ( b ) σ k ( c ) + σ 0 ( c ) σ k ( b ). Equating the degree k components from b oth sides of the equation gives: (5.1) σ k ( bc ) = σ 0 ( b ) σ k ( c ) + σ 0 ( c ) σ k ( b ) . Since A k is a free B -mo dule with basis M k , therefore σ k ( b ) can b e expressed uniquely in terms of this basis σ k ( b ) = P m ∈ M k D m ( b ) m , where each co efficien t map D m : B → B inherits the identit y from (5.1): D m ( bc ) = σ 0 ( b ) D m ( c ) + σ 0 ( c ) D m ( b ) . As σ is R -linear, we conclude that each co efficien t map D m is R -linear. The fact that D m = 0 is a natural extension of the standard result that commutativ e rings in whic h elemen ts ha v e roots admit no non-trivial deriv ations. F or the sak e of completeness, we include a self-contained proof, whic h relies only on the basic algebraic prop erties of B . Since D m (1) = 0, therefore D m ( β ) = 0 for all β ∈ R . Let g ∈ B and let y ∈ T . Also, let σ 0 ( g )( y ) = t and let F = g − t . It is easy to see that σ 0 ( F )( y ) = 0. If w e tak e G = F 1 3 and H = F 2 3 , then F = GH with σ 0 ( G )( y ) = 0 and σ 0 ( H )( y ) = 0. As D m ( F ) = σ 0 ( G ) D m ( H ) + σ 0 ( H ) D m ( G ), we obtain that D m ( F )( y ) = 0. F urther, since D m ( g ) = D m ( F ), w e conclude that D m ( g )( y ) = 0. Since y ∈ T w as arbitrary , we get that D m = 0. Because D m = 0 for all m ∈ M k , we conclude that σ k ( b ) = 0 for all b ∈ B . This contradicts the assumption that k is the minimal positive integer for which σ k is non-zero. This sho ws that σ ( b ) = σ 0 ( b ) ∈ B for all b ∈ B . Hence, B is fully inv ariant in A . No w, Theorem (4.3) implies that i ∗ : π alg R ( A ) → π alg R ( B ) is surjective, where i : B  → A is the inclusion. Since π alg R ( B )  = 0, w e conclude that π alg R ( A )  = 0. ■ References 1. L. Gillman and M. Jerison, Rings of Continuous F unctions , D. V an Nostrand, Princeton, NJ, (1960). 2. R. Bro wn, F unction sp ac es and pro duct topolo gies , Q. J. Math., V ol. 15 (1), pp. 238–250, (1964). 3. G. D. Marco, A. Orsatti, Commutative rings in which every prime ideal is c ontaine d in a unique maximal ide al , Proc. Amer. Math. So c., V ol. 30 (3), pp. 459–466, (1971). 4. M. Contessa, On pm-rings , Comm. Alg., V ol. 10 (1), pp. 93–108, (1982). Dep ar tment of Ma thema tics, The University of Burdw an, Burdw an Rajba ti, West Bengal 713104 Email address : bmitra@math.buruniv.ac.in Dep ar tment of Ma thema tics, The University of Burdw an, Burdw an Rajba ti, West Bengal 713104 Email address : harakrishnaranusourav@gmail.com