Localization of injective modules over valuation rings

It is proved that is injective if is an injective module over a valuation ring , for each prime ideal . Moreover, if or is flat, then is injective, too. It follows that localizations of injective modules over h-local Prüfer domains are injective, too.

     

Two characterizations of pure injective modules

Let be a commutative ring with identity and an -module. It is shown that if is pure injective, then is isomorphic to a direct summand of the direct product of a family of finitely embedded modules. As a result, it follows that if is Noetherian, then is pure injective if and only if is isomorphic to a direct summand of the direct product of a family of Artinian modules. Moreover, it is proved that is pure injective if and only if there is a family of -algebras which are finitely presented as -modules, such that is isomorphic to a direct summand of a module of the form , where for each , is an injective -module.

     

The rank of finitely generated modules over group algebras

We show the existence of a rank function on finitely generated modules over group algebras , where is an arbitrary field and is a finitely generated amenable group. This extends a result of W. Lück (1998).

     

Limits of Pure-Injective Cotilting Modules

The set of pure-injective cotilting modules over an artin algebra is shown to have a monoid structure. This monoid structure does not restrict down to a monoid structure on the finitely generated cotilting modules in general, but it does whenever the algebra is of finite representation type. Pure-injective cotilting modules are also constructed from any set of finitely generated cotilting modules with bounded injective dimension. Presented by Y. Drozd Mathematics Subject Classifications (2000) 16G10, 16P20, 16E30.     

Injective modules and semistar operations

We approach the problem of classifying injective modules over an integral domain, by considering the class of semistar Noetherian domains. When working with such domains, one has to focus on semistar ideals: as a consequence for modules, we restrict our study to the class of injective hulls of co-semistar modules, those in which the annihilator ideal of each nonzero element is semistar. We obtain a complete classification of this class, by describing its elements as injective hulls of uniquely determined direct sums of indecomposable injective modules; if moreover, we consider stable semistar operations, then we can further improve this result, obtaining a natural generalization of the classical Noetherian case. Our approach provides a unified treatment of results on injective modules over various kinds of domains obtained by Matlis, Cailleau, Beck, Fuchs and Kim–Kim–Park.     

Composition series of modules over Prüfer domains

A weakened version of the Jordan-Hölder theorem is shown to hold for torsion-free finite rank modules over an integral domain precisely when is a Prüfer domain.

     

Foxby duality and Gorenstein injective and projective modules

In 1966, Auslander introduced the notion of the -dimension of a finitely generated module over a Cohen-Macaulay noetherian ring and found the basic properties of these dimensions. His results were valid over a local Cohen-Macaulay ring admitting a dualizing module (also see Auslander and Bridger (Mem. Amer. Math. Soc., vol. 94, 1969)). Enochs and Jenda attempted to dualize the notion of -dimensions. It seemed appropriate to call the modules with -dimension 0 Gorenstein projective, so the basic problem was to define Gorenstein injective modules. These were defined in Math. Z. 220 (1995), 611--633 and were shown to have properties predicted by Auslander's results. The way we define Gorenstein injective modules can be dualized, and so we can define Gorenstein projective modules (i.e. modules of -dimension 0) whether the modules are finitely generated or not. The investigation of these modules and also Gorenstein flat modules was continued by Enochs, Jenda, Xu and Torrecillas. However, to get good results it was necessary to take the base ring Gorenstein. H.-B. Foxby introduced a duality between two full subcategories in the category of modules over a local Cohen-Macaulay ring admitting a dualizing module. He proved that the finitely generated modules in one category are precisely those of finite -dimension. We extend this result to modules which are not necessarily finitely generated and also prove the dual result, i.e. we characterize the modules in the other class defined by Foxby. The basic result of this paper is that the two classes involved in Foxby's duality coincide with the classes of those modules having finite Gorenstein projective and those having finite Gorenstein injective dimensions. We note that this duality then allows us to extend many of our results to the original Auslander setting.

     

On the closedness of taking injective hulls of several Serre subcategories

Belshoff and Xu showed that every Matlis reflexive module has a Matlis reflexive injective hull if and only if R is complete and has dimension less than or equal to 1. In this paper, we give a characterization of the closedness of taking injective hulls for a Serre subcategory consisting of Minimax modules. In addition, the closedness of taking injective hulls for a Serre subcategory consisting of extension modules of finitely generated modules by modules with finite support is characterized by the number of prime ideals. Our results provide a negative answer to Aghapournahr and Melkersson’s question concerning Melkersson subcategories.     

Pure-injective hulls of modules over valuation rings

If is the pure-injective hull of a valuation ring R, it is proved that is the pure-injective hull of M, for every finitely generated R-module M. Moreover , where (A_k)_1≤k≤n is the annihilator sequence of M. The pure-injective hulls of uniserial or polyserial modules are also investigated. Any two pure-composition series of a countably generated polyserial module are isomorphic.     

Cyclically Presented Modules Over Rings of Finite Type

A ring is of finite type if it has only finitely many maximal right ideals, all two-sided. In this article, we give a complete set of invariants for finite direct sums of cyclically presented modules over a ring R of finite type. More generally, our results apply to finite direct sums of direct summands of cyclically presented right R-modules (DCP modules). Using a duality, we obtain as an application a similar set of invariants for kernels of morphisms between finite direct sums of pair-wise non-isomorphic indecomposable injective modules over an arbitrary ring. This application motivates the study of DCP modules.     

Cobalanced Extensions of a Torsion-Free Module by a Torsion Module Over an Integral Domain

Cobalanced extensions of torsion-free modules by torsion modules over domains are investigated. As in the case of Abelian Groups, these sequences will split if and only if the torsion-free module is a pure submodule of a vector module. Unlike this case, even the torsion-free modules of finite rank need not be locally completely decomposable. For 1-dimensional Noetherian domains, necessary and sufficient conditions are given for that to be true.     

Selforthogonal modules with finite injective dimension

The category consisting of finitely generated modules which are left orthogonal with a cotilting bimodule is shown to be functorially finite. The notion of left orthogonal dimension is introduced, and then a necessary and sufficient condition of selforthogonal modules having finite injective dimension and a characterization of cotilting modules are given.     

Classifying Serre subcategories of finitely presented modules

Given a commutative coherent ring , a bijective correspondence between the thick subcategories of perfect complexes and the Serre subcategories of finitely presented modules is established. To construct this correspondence, properties of the Ziegler and Zariski topologies on the set of isomorphism classes of indecomposable injective modules are used in an essential way.

     

(Strongly) Gorenstein injective modules over upper triangular matrix Artin algebras

Let \(\Lambda = \left( {\begin{array}{*{20}{c}} A&M \\ 0&B \end{array}} \right)\) be an Artin algebra. In view of the characterization of finitely generated Gorenstein injective Λ-modules under the condition that M is a cocompatible (A,B)-bimodule, we establish a recollement of the stable category \(\overline {Ginj\left( \Lambda \right)} \). We also determine all strongly complete injective resolutions and all strongly Gorenstein injective modules over Λ.     

Torsions in modules

In the category of right modules over the ring E=End_R(F), where F is a free right R-module, a torsion is defined. It is known as Tol'skaya torsion. The correlation between torsion-free E-modules in the sense of Tol'skaya and torsion-free E-modules in the sense of Bass is investigated. It is shown that the ring R is a right cogenerator if and only if in the ring of endomorphisms of any free R-module, r((J))=J for all finitely generated right ideals J.Translated from Matematicheskie Zametki, Vol. 14, No. 4, pp. 527–534, October, 1973.The author thanks L. A. Skornyakov for his valuable discussions and observations.     

Decomposition of injective modules relative to a torsion theory

IfR is a right noetherian ring, the decomposition of an injective module, as a direct sum of uniform submodules, is well known. Also, this property characterises this kind of ring. M. L. Teply obtains this result for torsion-free injective modules. The decomposition of injective modules relative to a torsion theory has been studied by S. Mohamed, S. Singh, K. Masaike and T. Horigone. In this paper our aim is to determine those rings satisfying that every torsion-freeτ-injective module is a direct sum ofτ-uniformτ-injective submodules and also to determine those rings with the same property for everyτ-injective module.     

Strongly Flat Modules over Matlis Domains

Strongly flat modules were introduced by Bazzoni–Salce [3 Bazzoni , S. , Salce , L. ( 2003 ). Almost perfect domains . Colloq. Math. 95 : 285 – 301 .[Crossref] , [Google Scholar]] and used to characterize almost perfect domains. Here we wish to study strongly flat modules, more generally, over Matlis domains; these are integral domains R such that the field of quotients Q has projective dimension 1. In Section 2, criteria are proved for strong flatness. We also prove that over arbitrary domains, strongly flat submodules of projective modules are projective (Theorem 3.2), in particular, strongly flat ideals are projective (Corollary 3.4) and use these results to show that the strongly flat dimension (which makes sense over Matlis domains) coincides with the projective dimension whenever it is > 1.