Rings and modules characterized by opposites of injectivity

Abstract

In a recent paper, Aydoǧdu and López-Permouth have defined a module M to be N -subinjective if every homomorphism N → M extends to some E(N ) → M , where E(N ) is the injective hull of N . Clearly, every module is subinjective relative to any injective module. Their work raises the following question: What is the structure of a ring over which every module is injective or subinjective relative only to the smallest possible family of modules, namely injectives? We show, using a dual opposite injectivity condition, that such a ring R is isomorphic to the direct product of a semisimple Artinian ring and an indecomposable ring which is (i) a hereditary Artinian serial ring with J 2 = 0; or (ii) a QF-ring isomorphic to a matrix ring over a local ring. Each case is viable and, conversely, (i) is sufficient for the said property, and a partial converse is proved for a ring satisfying (ii). Using the above mentioned classification, it is also shown that such rings coincide with the fully saturated rings of Trlifaj except, possibly, when von Neumann regularity is assumed. Furthermore, rings and abelian groups which satisfy these opposite injectivity conditions are characterized.

Subprojectivity domains of pure-projective modules

“…finitely generated) modules are i-test. In [1,Theorem 16], authors proved that a non von Neumann regular ring R is right fully saturated if and only if all non-injective modules are indigent.…”

“…In [1], the authors interested with the structure of rings over which every noninjective module is indigent. In this section, we deal with the structure of rings over which every non-injective simple (respectively, singular, uniform, indecomposable) module is indigent.…”

“…Then R has a unique non-injective simple right R-module, say U . U is indigent by [1,Proposition 32].…”

“…Presently, it is not known whether indigent module exists for an arbitrary ring, but an affirmative answer is known for some rings, such as Noetherian rings [9]. In [1], the authors studied ring R with the property that every non-injective right module is indigent. Indigent modules have been recently studied in [2].…”

“…Finally, we study the structure of a ring R with the property that every noninjective simple right module is indigent. In [1], Alizade, Büyükaşik and Er also investigate when non injective simple modules are indigents. We improve their results proving that every non-injective simple right R-module is indigent if and only if (i) R is a right V-ring; or, (ii)R is right Hereditary righ Noetherian ring and, for any right R-module M , either M is indigent or Soc(M ) = Soc(N ), where N is the largest injective submodule of M ; or, (iii)R ∼ = S × T , where S is semisimple Artinian ring and T is an indecomposable matrix ring over a local QF-ring (Theorem 18).…”