On Artinian rings with restricted class of injectivity domains

AYDOĞDU P., SARAC B.

JOURNAL OF ALGEBRA, vol.377, pp.49-65, 2013 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 377
  • Publication Date: 2013
  • Doi Number: 10.1016/j.jalgebra.2012.11.027
  • Journal Name: JOURNAL OF ALGEBRA
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.49-65
  • Keywords: Injective modules, Poor modules, Injectivity domain, Artinian rings
  • Hacettepe University Affiliated: Yes

Abstract

In a recent paper of Alahmadi, Alkan and Lopez-Permouth, a ring R is defined to have no (simple) middle class if the injectivity domain of any (simple) R-module is the smallest or largest possible. Er, Lopez-Permouth and Sokmez use this idea of restricting the class of injectivity domains to classify rings, and give a partial characterization of rings with no middle class. In this work, we continue the study of the property of having no (simple) middle class. We give a structural description of right Artinian right nonsingular rings with no right middle class. We also give a characterization of right Artinian rings that are not SI to have no middle class, which gives rise to a full characterization of rings with no middle class. Furthermore, we show that commutative rings with no middle class are those Artinian rings which decompose into a sum of a semisimple ring and a ring of composition length two. Also, Artinian rings with no simple middle class are characterized. We demonstrate our results with several examples. (c) 2012 Elsevier Inc. All rights reserved.