The Z* functor for rings whose primitive images are Artinian

Ozcan A. Ç., SMITH P.

COMMUNICATIONS IN ALGEBRA, vol.30, no.10, pp.4915-4930, 2002 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 30 Issue: 10
  • Publication Date: 2002
  • Doi Number: 10.1081/agb-120014676
  • Journal Name: COMMUNICATIONS IN ALGEBRA
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.4915-4930
  • Hacettepe University Affiliated: Yes

Abstract

Given a ring R, we investigate a subfunctor Z* of the identity functor on the category of all right R-modules which is defined by Z* (M) = {m is an element of M: mR is a small module}, for any R-module M. We prove that if the ring R satisfies the descending chain condition for right annihilators and RIP is an Artinian ring for every primitive ideal P then Z*(M) = {m is an element of M: mS = 0} for every right R-module M, where S is the left socle of R. Moreover the ring R is semiprime Artinian if and only if R is right bounded, R satisfies the descending chain condition for right annihilators and Z*(M) = 0 for some faithful right R-module M.