Gorenstein injective and strongly cotorsion modules

By investigating the properties of some special covers and envelopes of modules, we prove that if R is a Gorenstein ring with the injective envelope of _R R flat, then a left R-module is Gorenstein injective if and only if it is strongly cotorsion, and a right R-module is Gorenstein flat if and only if it is strongly torsionfree. As a consequence, we get that for an Auslander-Gorenstein ring R, a left R-module is Gorenstein injective (resp. flat) if and only if it is strongly cotorsion (resp. torsionfree).     

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.

     

Gorenstein injective modules and Auslander categories

In this paper, we study Gorenstein injective modules over a local Noetherian ring R. For an R-module M, we show that M is Gorenstein injective if and only if Hom _R (Ȓ,M) belongs to Auslander category B(Ȓ), M is cotorsion and Ext ⁱ _R (E,M) = 0 for all injective R-modules E and all i > 0. Received: 24 August 2006 Revised: 30 October 2006     

Extending modules which are direct sums of injective modules and semisimple modules

It is proved that, if R is a right Noetherian ringM ₁ is an injective right R-module and M ₂ is a semisimple right R-module, then the right R-module M ₁ + M ₂ is extending if and only if M ₂ is (M ₁/Soc(M ₁))-injective.     

The flat dimensions of injective modules

Let R be a ring. Define r. IFD(R) as r.IFD(R)=sup{fdE/E is an injective right R—module}. The purpose of this paper is to investigate this “global” dimension.     

Gorenstein injective modules and local cohomology

In this paper we assume that is a Gorenstein Noetherian ring. We show that if is also a local ring with Krull dimension that is less than or equal to 2, then for any nonzero ideal of , is Gorenstein injective. We establish a relation between Gorenstein injective modules and local cohomology. In fact, we will show that if is a Gorenstein ring, then for any -module its local cohomology modules can be calculated by means of a resolution of by Gorenstein injective modules. Also we prove that if is -Gorenstein, is a Gorenstein injective and is a nonzero ideal of , then is Gorenstein injective.

     

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.     

Relative copure injective and copure flat modules

Let R be a ring, n a fixed nonnegative integer and I_n (F_n) the class of all left (right) R-modules of injective (flat) dimension at most n. A left R-module M (resp., right R-module F) is called n-copure injective (resp., n-copure flat) if (resp., ) for any N∈I_n. It is shown that a left R-module M over any ring R is n-copure injective if and only if M is a kernel of an I_n-precover f:A→B of a left R-module B with A injective. For a left coherent ring R, it is proven that every right R-module has an F_n-preenvelope, and a finitely presented right R-module M is n-copure flat if and only if M is a cokernel of an F_n-preenvelope K→F of a right R-module K with F flat. These classes of modules are also used to construct cotorsion theories and to characterize the global dimension of a ring under suitable conditions.     

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.