Nilpotent-invariant modules and rings

Kosan M. T., Truong Cong Quynh T. C. Q.

COMMUNICATIONS IN ALGEBRA, vol.45, no.7, pp.2775-2782, 2017 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 45 Issue: 7
  • Publication Date: 2017
  • Doi Number: 10.1080/00927872.2016.1226873
  • Journal Name: COMMUNICATIONS IN ALGEBRA
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.2775-2782
  • Keywords: Automorphism-invariant, module, nilpotent endomorphism, nilpotent-invariant module
  • Gazi University Affiliated: No

Abstract

Automorphism-invariant modules, due to Lee and Zhou, generalize the notion of quasi-injective modules. A module which is invariant under automorphisms of its injective hull is called an automorphism-invariant module. Here we carry out a study of the module which is invariant under nilpotent endomorphisms of its injective envelope, such as modules are called nilpotent-invariant. Many basic properties are obtained. For instance, it is proved that (1) nilpotent-invariant modules have the (C3) property, (2) if M = M-1 circle plus M-2 is nilpotent-invariant, then M-1 and M-2 are relative injective. In this paper, we also show that (3) a simple right nilpotent-invariant ring R is either right self-injective or R-R is uniform square-free.