Weak Injective Modules Research Articles

When weak-injective modules decompose like injectives

When weak-injective modules decompose like injectives

Remarks on Gorenstein Weak Injective and Weak Flat Modules

In this paper, we introduce Gorenstein weak injective and weak flat modules in terms of, respectively, weak injective and weak flat modules; the classes of Gorenstein weak injective and weak flat modules are larger than the classical classes of Gorenstein injective and flat modules. In this new setting, we characterize rings over which all modules are Gorenstein weak injective. Moreover, we discuss the relation between the weak cosyzygy and Gorenstein weak cosyzygy of a module, and also the stability of Gorenstein weak injective modules.

Weak-Injective Modules of Weak Dimension at Most One

We consider modules over integral domains R. A main purpose is to show that certain module properties assumed on R-modules of weak dimension ≤1 imply that these properties are shared by all modules in the category of R-modules. Also we prove several results involving modules of weak dimension ≤1.

Weak-projective dimensions

In this paper, the notions of weak-projective modules and weak-projective dimension over commutative domain R are given. It is shown that over semisimple rings with weak global dimension 1, these modules are equivalent to weak-injective modules. The weak-projective dimension measures how far away a domain is from being a Prüfer domain. Several properties of these modules are also presented.

DIVISIBLE ENVELOPES, $\mathcal{P}_1$-COVERS AND WEAK-INJECTIVE MODULES

We consider the problem of characterizing the commutative domains such that every module admits a divisible envelope or a [Formula: see text]-cover, where [Formula: see text] is the class of modules of projective dimension at most one. We prove that each one of the two conditions is satisfied exactly for the class of almost perfect domains, that is commutative domains such that all of their proper quotients are perfect rings. Moreover, we show that the class of weak-injective modules over a commutative domain is closed under direct sums if and only if the domain is almost perfect.

Weak-injectivity and almost perfect domains

Our main purpose is to characterize the class of almost perfect domains (introduced by Bazzoni and Salce [S. Bazzoni, L. Salce, Almost perfect domains, Colloq. Math. 95 (2003) 285–301]) by weak-injectivity. Weak-injective modules have been discussed by Lee [S.B. Lee, Weak-injective modules, Comm. Algebra 34 (2006) 361–370; S.B. Lee, A note on the Matlis category equivalence, J. Algebra 299 (2006) 854–862]. Here we add some more results on weak-injective modules before we give several characterizations of almost perfect domains.

Weak-Injective Modules

ABSTRACT Pure-injective and RD-injective R-modules over domains R have been investigated by many authors. We introduce another class of R-modules, called weak-injective modules, which turn out to be useful in addressing several unanswered questions between the two classes of modules. We also find that this class is an envelope class over any domain, giving a partial answer to the existence of envelope classes in the hierarchy of injective and divisible modules. Communicated by I. Swanson.

A note on the Matlis category equivalence

We provide additional information concerning the important Matlis category equivalence between the categories of h-divisible torsion and R-complete torsion-free R-modules over any domain R. We identify the h-divisible modules that correspond under this category equivalence to the Enochs cotorsion torsion-free modules as the weak-injective modules. As a consequence, we can characterize weak-injective modules in terms of their flat covers. We also determine the range of the injective dimensions of pure-injective modules.

Weakly-injective modules over hereditary noetherian prime rings

AbstractA module M is said to be wealdy-injective if and only if for every finitely generated submodule N of the injective hull E(M) of M there exists a submodule X of E(M), isomorphic to M such that N ⊂ X. In this paper we investigate weakly-injective modules over bounded hereditary noetherian prime rings. In particular we show that torsion-free modules over bounded hnp rings are always wealdy-injective, while torsion modules with finite Goldie dimension are weakly-injective only if they are injective.As an application, we show that weakly-injective modules over bounded Dedekind prime rings have a decomposition as a direct sum of an injective module B, and a module C satisfying that if a simple module S is embeddable in C then the (external) direct sum of all proper submodules of the injective hull of S is also embeddable in C. Indeed, we show that over a bounded hereditary noetherian prime ring every uniform module has periodicity one if and only if every weakly-injective module has such a decomposition.

Strongly compressible modules and semiprime right goldie rings

A module M R is defined to be strongly compressible (or SC for short) if for every essential submodule N of M, there exists X ⊆ E(M) such that M ⊆ X ≅ N. We show that a ring R is semiprime right Goldie iff R Ris SC module iff every right ideal of R is SC module iff every submodule of each progenerator of Mod-R is SC module. As corollaries of this result, we obtain some new module-theoretic characterizations of semiprime Goldie rings, prime (right) Goldie rings and Prüfer rings, etc., etc.,respectively. And the characterization theorem of semiprime Goldie rings of López-Permouth, Rizvi and Yousif by using weakly-injective modules can be regarded as a corollary of our results.

Rings characterized by their weakly-injective modules

The notation in this paper will be standard and it may be found in [2] or [8]. Throughout the paper, the notation A ⊂' B will mean that A is an essential submodule of the module B. Given an arbitrary ring R and R-modules M and N, we say that M is weakly N-injective if and only if every map φ:N → E(M) from N into the injective hull E(M) of M may be written as a composition σ〫 , where :N→M and σ:M→E(M) is a monomorphism. This is equivalent to saying that for every map φ:N→E(M), there exists a submodule X of E(M), isomorphic to M, such that φ(N) is contained in X. In particular, M is weakly R-injective if and only if, for every x ∈ E(M), there exists X ⊂ E(M) such that x ∈ X ≌ M. We say that M is weakly-injective if and only if it is weakly N-innjective for every finitely generated module N. Clearly, M is weakly-injective if and only if, for every finitely generated submodule N of E(M), there exists X ⊂ E(M) such that N ⊂ X ≌ M.