Noetherian rings with almost injective simple modules

We characterize right Noetherian rings over which all simple modules are almost injective. It is proved that R is such a ring, if and only if, the complements of semisimple submodules of every R-module M are direct summands of M, if and only if, R is a finite direct sum of right ideals I_r, where I_r is either a Noetherian V-module with zero socle, or a simple module, or an injective module of length 2. A commutative Noetherian ring for which all simple modules are almost injective is precisely a finite direct product of rings R_i, where R_i is either a field or a quasi-Frobenius ring of length 2. We show that for commutative rings whose all simple modules are almost injective, the properties of Kasch, (semi)perfect, semilocal, quasi-Frobenius, Artinian, and Noetherian coincide.     

Direct injective modules

Throughout this paperR will denote a ring with idenity element andM a unitary right module overR. AnR-moduleM is said to be direct injective if and only if given direct summandN ofM with injectioni _N:N→M and a monomorphismg:N→M, there exists an endomorphismf ofR-moduleM such thatfg=i _N. In this paper we investigate properties of direct injective modules, and obtain the following results on direct injective modules.

1. We establish the necessary and sufficient condition for a module to be direct injective.
2. We show that the answer on problem of Krull-Schmidt-Matlis is in the affirmative in caseR-moduleM is extending direct injective.
3. We prove that extending direct injectivity of module implies same properties of its direct summands.

     

On the structure of indecomposable injective modules 1

For a ring R, the following two conditions are equivalent:.

(1) If E is an indecomposable injective right R-module, then End E_R is a field (not necesarily commutative).

(2) Every co-irreducible rigtht ideal is critical.

Since (2) has been characterized ideal-theoretically, this amounts to an ideal-theoretical characterization of (1). These rings come up to the study of (QI) rings in which every quasi-injective module is injective.     

On Gorenstein injective discrete modules over profinite groups

Gorenstein homological algebra was introduced in categories of modules. But it has proved to be a fruitful way to study various other categories such as categories of complexes and of sheaves. In this paper, the research of relative homological algebra in categories of discrete modules over profinite groups is initiated. This seems appropriate since (in some sense) the subject of Gorenstein homological algebra had its beginning with Tate homology and cohomology over finite groups. We prove that if the profinite group has virtually finite cohomological dimension then every discrete module has a Gorenstein injective envelope, a Gorenstein injective cover and we study various cohomological dimensions relative to Gorenstein injective discrete modules. We also study the connection between relative and Tate cohomology theories.     

On injective modules over Azumaya algebras over locally noetherian schemes

Let be an Azumaya algebra over a locally noetherian scheme X. We describe in this work quasi-coherent -bimodules which are injective in the category of sheaves of left -modules     

On finitely injective modules and locally pure-injective modules over Prüfer domains

Over Matlis valuation domains there exist finitely injective modules which are not direct sums of injective modules, as well as complete locally pure-injective modules which are not the completion of a direct sum of pure-injective modules. Over Prüfer domains which are either almost maximal, or -local Matlis, finitely injective torsion modules and complete torsion-free locally pure-injective modules correspond to each other under the Matlis equivalence. Almost maximal Prüfer domains are characterized by the property that every torsion-free complete module is locally pure-injective. It is derived that semi-Dedekind domains are Dedekind.

     

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).