Indecomposable Injective R-modules Research Articles

Injective Modules over Prüfer v-Multiplication Domains

Let R be an integral domain with quotient field F. It is shown that R is a strongly discrete Prüfer v-multiplication domain if and only if there exists a bijection between the set of the prime w-ideals and the set of isomorphism classes of GV-torsionfree indecomposable injective R-modules and every indecomposable injective R-module, viewed as a module over its endomorphism ring, is uniserial. It is also shown that the w-closure of any GV-torsionfree homomorphic image of F is injective if and only if R is a Prüfer v-multiplication domain satisfying an almost maximality-type property.

Gorenstein Injective Dimension and Dual Auslander–Bridger Formula#

An R-module M satisfies the dual Auslander–Bridger formula if Gid R M + Tor − depth M = depth R. Let R be a complete Cohen–Macaulay local ring with residue field k and M be a non-injective Ext-finite R-module such that for all i ≥ 1 and all indecomposable injective R-modules E ≠ E(k). If M has finite Gorenstein injective dimension, then we will prove that M satisfies the dual Auslander–Bridger formula if either Ext-depth M > 0 or Ext-depth M = 0 and Gid R M ≠ 1. We denote Gorenstein injective envelope of M by G(M). If R is a Gorenstein local ring and M is a non-Gorenstein injective finitely generated R-module and G(M) is reduced, then this formula holds for and its cosyzygy. As the last result, if R is a regular local ring, then dual Auslander–Bridger holds for any non-injective Ext-finite R − module.

Injective modules and fp-injective modules over valuation rings

It is shown that each almost maximal valuation ring R, such that every indecomposable injective R-module is countably generated, satisfies the following condition (C): each fp-injective R-module is locally injective. The converse holds if R is a domain. Moreover, it is proved that a valuation ring R that satisfies this condition (C) is almost maximal. The converse holds if Spec( R) is countable. When this last condition is satisfied it is also proved that every ideal of R is countably generated. New criteria for a valuation ring to be almost maximal are given. They generalize the criterion given by E. Matlis in the domain case. Necessary and sufficient conditions for a valuation ring to be an IF-ring are also given.

Gorenstein injective dimension and Tor-depth of modules

The main aim of this paper is to obtain a dual result to the now well known Auslander-Bridger formula for G-dimension. We will show that if R is a complete Cohen-Macaulay ring with residue field k, and M is a non-injective h-divisible Ext-finite R-module of finite Gorenstein injective dimension such that for each $i \geq 1$ Ext i (E,M) = 0 for all indecomposable injective R-modules $E \neq E(k)$ , then the depth of the ring is equal to the sum of the Gorenstein injective dimension and Tor-depth of M. As a consequence, we get that this formula holds over a d-dimensional Gorenstein local ring for every nonzero cosyzygy of a finitely generated R-module and thus in particular each such n th cosyzygy has its Tor-depth equal to the depth of the ring whenever $n \geq d$ .

Generalized Dedekind domains and their injective modules

We prove that for a commutative integral domain R the following conditions are equivalent: (a) R is a Prüfer domain with no non-zero idempotent prime ideals; (b) there is a one to one correspondence between prime ideals in R and isomorphism classes of indecomposable injective R-modules, and every indecomposable injective R-module, viewed as a module over its endomorphism ring, is uniserial. This result allows us to study and describe injective modules over generalized Dedekind domains. Furthermore, we show that a partially ordered set is order isomorphic to the spectrum of a generalized Dedekind domain if and only if it is a Noetherian tree with a least element.