Please use this identifier to cite or link to this item: https://hdl.handle.net/11147/5422
Title: Rings and modules characterized by opposites of injectivity
Authors: Alizade, Rafail
Büyükaşık, Engin
Er, Noyan
Keywords: Artinian serial
Fully saturated
Injective
Subinjective
QF ring
Issue Date: Jul-2014
Publisher: Academic Press Inc.
Source: Alizade, R., Büyükaşik, E., and Er, N. (2014). Rings and modules characterized by opposites of injectivity. Journal of Algebra, 409, 182-198. doi:10.1016/j.jalgebra.2014.03.027
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.
URI: http://doi.org/10.1016/j.jalgebra.2014.03.027
http://hdl.handle.net/11147/5422
ISSN: 0021-8693
1090-266X
Appears in Collections:Mathematics / Matematik
Scopus İndeksli Yayınlar Koleksiyonu / Scopus Indexed Publications Collection
WoS İndeksli Yayınlar Koleksiyonu / WoS Indexed Publications Collection

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