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 J2 = 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. © 2014 Elsevier Inc. © 2014 Elsevier B.V. All rights reserved.