A duality principle for lattices and categories of modules

Let R be a ring with 1, R^op the opposite ring, and R-Mod the category of left unitary R-modules and R-linear maps. A characterization of well-powered abelian categories $\text{A}$ such that there exists an exact embedding functor $\text{A}$→R-Mod is given. Using this characterization and abelian category duality, the following duality principles can be established.Theorem. There exists an exact embedding functor $\text{A}$→R-Mod if and only if there exists an exact embedding functor $\text{A}$^op→R^op-Mod.Corollary. If R-Mod has a specified diagram-chasing property, then R^op-Mod has the dual property.A lattice L is representable by R-modules if it is embeddable in the lattice of submodules of some unitary left R-module; $\text{L}$(R) denotes the quasivariety of all lattices representable by R-modules.Theorem. A lattice L is representable by R-modules if and only if its order dual L¹ is representable by R^op-modules. That is, $\text{L}\text{(}\text{R}{}^{\mathrm{op}}\text{)={L}{}^1\text{:L?}\text{L}\text{(}\text{R}\text{)}}$.If $\text{R}$ is a commutative ring with 1 and a specified diagram-chasing result is satisfied in R-Mod, then the dual result is also satisfied in R-Mod. Furthermore, $\text{L}\text{(}\text{R}\text{)}$ is self-dual: $\text{L}\text{(}\text{R}\text{)= {L}{}^1\text{:L?}\text{L}\text{(}\text{R}\text{)}.}$     

On Divisible and Torsionfree Modules

A ring R is called left P-coherent in case each principal left ideal of R is finitely presented. A left R-module M (resp. right R-module N) is called D-injective (resp. D-flat) if Ext¹(G, M) = 0 (resp. Tor₁(N, G) = 0) for every divisible left R-module G. It is shown that every left R-module over a left P-coherent ring R has a divisible cover; a left R-module M is D-injective if and only if M is the kernel of a divisible precover A → B with A injective; a finitely presented right R-module L over a left P-coherent ring R is D-flat if and only if L is the cokernel of a torsionfree preenvelope K → F with F flat. We also study the divisible and torsionfree dimensions of modules and rings. As applications, some new characterizations of von Neumann regular rings and PP rings are given.     

Gorenstein Modules and Dualizing Modules

In this article, we study the characterizations of Gorenstein injective left S-modules and finitely generated Gorenstein projective left R-modules when there is a dualizing S-R-bimodule associated with a right noetherian ring R and a left noetherian ring S.     

Finitely Generated Flat Modules and a Characterization of Semiperfect Rings

Abstract

For a ring S, let K ₀(FGFl(S)) and K ₀(FGPr(S)) denote the Grothendieck groups of the category of all finitely generated flat S-modules and the category of all finitely generated projective S-modules respectively. We prove that a semilocal ring Ris semiperfect if and only if the group homomorphism K ₀(FGFl(R)) → K ₀(FGFl(R/J(R))) is an epimorphism and K ₀(FGFl(R)) = K ₀(FGPr(R)).     

The State Space of K 0 of Exchange Rings

ABSTRACT

For a directly finite exchange ring R which satisfies general comparability, we construct all extreme points of the state space S(V(R),? R?), where V(R) denotes the monoid of all isomorphic classes of finitely generated projective R-modules. From this, we further prove that S((K ₀(R),[R])) is affinely homeomorphic to M ₁ ⁺(BS(R)), where BS(R) denotes the spectrum of the Boolean algebra B(R) of all central idempotents in R, and M ⁺ ₁(BS(R)) the set of all probability measures on BS(R). These generalize the corresponding results on regular rings. Particularly, all of our results hold for exchange rings with all the idempotents central.     

Sulla funtorialità dell'inviluppoF

Riassunto SiaA un anello associativo unitario eF una topologia di Gabriel suA. Vengono date delle condizioni necessarie e sufficienti affinchè la funzione che ad ogniA-moduloM assegna l'inviluppoF-iniettivo{itE}{inF}({itM}) sia un funtore.

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Summary LetA be an associative ring with unity,F a Gabriel topology of left ideals onA andw the hereditary torsion class associated withF, on the categoryA-Mod of unitary leftA-modules. The autor prove the equivalence of the following statements:a) all leftA-modules areF;b) the full subcategory ofA-Mod of allFs modules is reflective;c) there exist a functor {itE}{inF}:A-Mod→A-Mod such that, for everyA-moduleM, {itE}{inF} is theF envelope ofM, and for every homomorphismf: M→N the diagram {fx113-1} is commutative.

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Lavoro eseguito nell'ambito dell'attività di ricerca del G.N.S.A.G.A. (C.N.R.).     

Separativity of Regular Rings in Which 2 Is Invertible

A regular ring R is separative provided that for all finitely generated projective right R-modules A and B, A⊕ A? A⊕ B? A⊕ B implies that A? B. We prove, in this article, that a regular ring R in which 2 is invertible is separative if and only if each a ∈ R satisfying R(1 ? a ²)R = Rr(a) = ?(a)R and i(End _R (aR)) = ∞ is unit-regular if and only if each a ∈ R satisfying R(1 ? a ²)R ∩ RaR = Rr(a) ∩ ?(a)R ∩ RaR and i(End _R (aR)) = ∞ is unit-regular. Further equivalent characterizations of such regular rings are also obtained.     

On categories associated to a Quasi-Hopf algebra

A quasi-Hopf algebra H can be seen as a commutative algebra A in the center 𝒵(H-Mod) of H-Mod. We show that the category of A-modules in 𝒵(H-Mod) is equivalent (as a monoidal category) to H-Mod. This can be regarded as a generalization of the structure theorem of Hopf bimodules of a Hopf algebra to the quasi-Hopf setting.     

Comparing First Order Theories of Modules over Group Rings II: Decidability

We consider R‐torsionfree modules over group rings RG, where R is a Dedekind domain and G is a finite group. In the first part of the paper [4] we compared the theory T(RG) of all R‐torsionfree RG‐modules and the theory T₀(RG) of RG‐lattices (i. e. finitely generated R‐torsionfree RG‐modules), and we realized that they are almost always different. Now we compare their behaviour with respect to decidability, when RG‐lattices are of finite, or wild representation type.     

DOI-KOPPINEN HOPF BIMODULES ARE MODULES

I construct a generalized twisted smash product A ^H B, which gives an abstract structure of Cibils-Rosso's algebra X associated to a finite-dimensional Hopf algebra H, for the H-bimodule algebra A and H-bicomodule algebra B. I show that the Doi-Koppinen Hopf (H, B, D)-bimodules are modules over a certain algebra which is of this type. Moreover, if D is finitely generated projective as a k-module, there exists a k-module-preserving equivalence of categories between the category of Doi-Koppinen (H, B, D)-Hopf bimodules and the category of left (D ^*op ? D ^*) ^{H?H op} (B ? B ^op )-modules.     

Properties of Modules and Rings Relative to Some Matrices

Let R be a ring and  _β×α(R) (?   _β×α(R)) the set of all β × α full (row finite) matrices over R where α and β ≥ 1 are two cardinal numbers. A left R-module M is said to be “injective relative” to a matrix A ? ?   _β×α(R) if every R-homomorphism from R ^(β) A to M extends to one from R ^(α) to M. It is proved that M is injective relative to A if and only if it is A-pure in every module which contains M as a submodule. A right R-module N is called flat relative to a matrix A ?  _β×α(R) if the canonical map μ: N? R ^(β) A → N ^α is a monomorphism. This extends the notion of (m, n)-flat modules so that n-projectivity, finitely projectivity, and τ-flatness can be redefined in terms of flatness relative to certain matrices. R is called left coherent relative to a matrix A ?  _β×α(R) if R ^(β) A is a left R-ML module. Some results on τ-coherent rings and (m, n)-coherent rings are extended.     

Indecomposable finitely generated modules over valuation domains

Summary LetR be a valuation domain,S a maximal immediate extension ofR. We introduce the definition of unitary independence. We use units ofS, which are unitarily independent over an ideal ofR, to construct indecomposable finitely generatedR-modules with Goldie dimension greater than one. We prove that, ifR is archimedean, the endomorphism ring of an indecomposable finitely generatedR-module is local. On the other hand, we prove that, ifR is a suitable non archimedean valuation domain, there exist indecomposable finitely generatedR-modulesM such that End (M) is not local.

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Riassunto SiaR un dominio di valutazione,S un'estensione massimale immediate diR. Si introduce la definizione di indipendenza unitaria. Si usano unità diS unitariamente indipendenti su un ideale diR per costruireR-moduli finitamente generati indecomponibili con dimensione di Goldie maggiore di uno. Si dimostra che, seR è archimedeo, l'anello degli endomorfismi di unR-modulo finitamente generato indecomponibile è locale. Si prova altresì che, seR è un opportuno dominio di valutazione non archimedeo, esistonoR-moduliM finitamente generati indecomponibili, tali che End (M) non è locale.

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Lavoro eseguito nell'ambito del GNSAGA.     

Derived Equivalences of Triangular Matrix Rings Arising from Extensions of Tilting Modules

A triangular matrix ring Λ is defined by a triplet (R,S,M) where R and S are rings and _R M _S is an S-R-bimodule. In the main theorem of this paper we show that if T _S is a tilting S-module, then under certain homological conditions on the S-module M _S , one can extend T _S to a tilting complex over Λ inducing a derived equivalence between Λ and another triangular matrix ring specified by (S′, R, M′), where the ring S′ and the R-S′-bimodule M′ depend only on M and T _S , and S′ is derived equivalent to S. Note that no conditions on the ring R are needed. These conditions are satisfied when S is an Artin algebra of finite global dimension and M _S is finitely generated. In this case, (S′,R,M′) = (S, R, DM) where D is the duality on the category of finitely generated S-modules. They are also satisfied when S is arbitrary, M _S has a finite projective resolution and Ext _S ⁿ (M _S , S) = 0 for all n > 0. In this case, (S′,R,M′) = (S, R, Hom _S (M, S)).     

Multiplication Modules in Which Every Prime Submodule is Contained in a Unique Maximal Submodule

Abstract

Let R be a commutative ring. An R-module M is called a multiplication module if for each submodule N of M, N?=?IM for some ideal I of R. An R-module M is called a pm-module, i.e., M is pm, if every prime submodule of M is contained in a unique maximal submodule of M. In this paper the following results are obtained. (1) If R is pm, then any multiplication R-module M is pm. (2) If M is finitely generated, then M is a multiplication module if and only if Spec(M) is a spectral space if and only if Spec(M)?=?{PM?|?P?∈?Spec(R) and P???M ^⊥}. (3) If M is a finitely generated multiplication R-module, then: (i) M is pm if and only if Max(M) is a retract of Spec(M) if and only if Spec(M) is normal if and only if M is a weakly Gelfand module; (ii) M is a Gelfand module if and only if Mod(M) is normal. (4) If M is a multiplication R-module, then Spec(M) is normal if and only if Mod(M) is weakly normal.     

On Galois Comodules

Generalizing the notion of Galois corings, Galois comodules were introduced as comodules P over an A-coring 𝒞 for which P _A is finitely generated and projective and the evaluation map μ𝒞:Hom ^𝒞 (P, 𝒞) ?  _S P → 𝒞 is an isomorphism (of corings) where S = End ^𝒞 (P). It has been observed that for such comodules the functors ? ?  _A 𝒞 and Hom _A (P, ?) ?  _S P from the category of right A-modules to the category of right 𝒞-comodules are isomorphic. In this note we use this isomorphism related to a comodule P to define Galois comodules without requiring P _A to be finitely generated and projective. This generalises the old notion with this name but we show that essential properties and relationships are maintained. Galois comodules are close to being generators and have common properties with tilting (co)modules. Some of our results also apply to generalised Hopf Galois (coalgebra Galois) extensions.     

Characterization of Quasi-Coherent Modules that are Module Schemes

Let R be a commutative ring with unit, and let E be an R-module. We say the functor of R-modules E, defined by E(B) = E ? _R B, is a quasi-coherent R-module, and its dual E* is an R-module scheme. Both types of R-module functors are essential for the development of the theory of the linear representations of an affine R-group. We prove that a quasi-coherent R-module E is an R-module scheme if and only if E is a projective R-module of finite type, and, as a consequence, we also characterize finitely generated projective R-modules.     

On Gorenstein Projective Dimensions of Unbounded Complexes

Let R → S be a ring homomorphism and X be a complex of R-modules. Then the complex of S-modules S?_R~L X in the derived category D(S) is constructed in the natural way. This paper is devoted to dealing with the relationships of the Gorenstein projective dimension of an R-complex X(possibly unbounded) with those of the S-complex S?_R~L X.It is shown that if R is a Noetherian ring of finite Krull dimension and φ : R → S is a faithfully flat ring homomorphism, then for any homologically degree-wise finite complex X, there is an equality Gpd_RX = GpdS(S?_R~L X). Similar result is obtained for Ding projective dimension of the S-complex S?_R~L X.     

Wakamatsu tilting modules with finite injective dimension

Let R be a left Noetherian ring, S a right Noetherian ring and _R ω a Wakamatsu tilting module with S = End( _R ω). We introduce the notion of the ω-torsionfree dimension of finitely generated R-modules and give some criteria for computing it. For any n ? 0, we prove that l.id _R (ω) = r.id _S (ω) ? n if and only if every finitely generated left R-module and every finitely generated right S-module have ω-torsionfree dimension at most n, if and only if every finitely generated left R-module (or right S-module) has generalized Gorenstein dimension at most n. Then some examples and applications are given.     

Estimates of global dimension

In this note we show that for a *ⁿ-module, in particular, an almost n-tilting module, P over a ring R with A = End_R P such that P _A has finite flat dimension, the upper bound of the global dimension of A can be estimated by the global dimension of R and hence generalize the corresponding results in tilting theory and the ones in the theory of *-modules. As an application, we show that for a finitely generated projective module over a VN regular ring R, the global dimension of its endomorphism ring is not more than the global dimension of R.