On (weakly) co-Hopfian automorphism-invariant modules

Quynh T. C., Abyzov A., Kosan M. T.

COMMUNICATIONS IN ALGEBRA, vol.48, no.7, pp.2894-2904, 2020 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 48 Issue: 7
  • Publication Date: 2020
  • Doi Number: 10.1080/00927872.2020.1723613
  • Journal Name: COMMUNICATIONS IN ALGEBRA
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Aerospace Database, Communication Abstracts, MathSciNet, Metadex, zbMATH, Civil Engineering Abstracts
  • Page Numbers: pp.2894-2904
  • Keywords: Automorphism-invariant modules, (weakly) Co-Hopfian modules, RIGHT IDEAL, RINGS, CANCELLATION
  • Gazi University Affiliated: Yes

Abstract

A module M over a ring R is called automorphism-invariant if M is invariant under automorphisms of its injective hull E(M) and M is called co-Hopfian if each injective endomorphism of M is an automorphism. It is shown that (1) being co-Hopfian, directly-finite and having the cancelation property or the the substitution property are all equivalent conditions on automorphism-invariant modules, (2) if is an automorphism-invariant module, then M is co-Hopfian iff M-1 and M-2 are co-Hopfian, (3) if M is an automorphism-invariant module, then M is co-Hopfian if and only if E(M) is co-Hopfian. The module M is called weakly co-Hopfian if any injective endomorphism of M is essential. We also show that (4) if R is a right and left automorphism-invariant ring, then R is stably finite iff R-R or is a co-Hopfian module iff if R-n is weakly co-Hopfian as a right or left R-module for all n is an element of N.