RINGS CHARACTERIZED VIA A CLASS OF LEFT EXACT PRERADICALS


ER N. F.

PROCEEDINGS OF THE EDINBURGH MATHEMATICAL SOCIETY, vol.59, no.3, pp.641-653, 2016 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 59 Issue: 3
  • Publication Date: 2016
  • Doi Number: 10.1017/s0013091515000206
  • Journal Name: PROCEEDINGS OF THE EDINBURGH MATHEMATICAL SOCIETY
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.641-653
  • Keywords: Ring, preradical, injective, MODULES, INJECTIVITY
  • Dokuz Eylül University Affiliated: Yes

Abstract

For two modules M and N, (i)(M)(N) stands for the largest submodule of N relative to which M is injective. For any module M, (i)(M) : Mod-R -> Mod-R thus defines a left exact preradical, and (i)(M)(M) is quasi-injective. Classes of ring including strongly prime, semi-Artinian rings and those with no middle class are characterized using this functor: a ring R is semi-simple or right strongly prime if and only if for any right R-module M, (i)(M)(R) - R or 0, extending a result of Rubin; R is a right QI-ring if and only if R has the ascending chain condition (a.c.c.) on essential right ideals and (i)(M) is a radical for each M is an element of Mod-R (the a.c.c. is not redundant), extending a partial answer of Dauns and Zhou to a long-standing open problem. Also discussed are rings close to those with no middle class.