Semisimple-direct-injective modules


Abyzov A., KOŞAN M. T. , Truong Cong Quynh T. C. Q. , Tapkin D.

HACETTEPE JOURNAL OF MATHEMATICS AND STATISTICS, vol.50, no.2, pp.516-525, 2021 (Journal Indexed in SCI) identifier

  • Publication Type: Article / Article
  • Volume: 50 Issue: 2
  • Publication Date: 2021
  • Doi Number: 10.15672/hujms.730907
  • Title of Journal : HACETTEPE JOURNAL OF MATHEMATICS AND STATISTICS
  • Page Numbers: pp.516-525

Abstract

The notion of simple-direct-injective modules which are a generalization of injective modules unifies C2 and C3-modules. In the present paper, we introduce the notion of the semisimple-direct-injective module which gives a unified viewpoint of C2, C3, SSP properties and simple-direct-injective modules. It is proved that a ring R is Artinian serial with the Jacobson radical square zero if and only if every semisimple-direct-injective right R-module has the SSP and, for any family of simple injective right R-modules {S-i}(J), circle plus S-J(i) is injective. We also show that R is a right Noetherian right V-ring if and only if every right R-module has a semisimple-direct-injective envelope if and only if every right R-module has a semisimple-direct-injective cover.