Modules with Finitely Many Submodules

Akbari S., GHEZELAHMAD S. K., Yaraneri E.

ALGEBRA COLLOQUIUM, vol.23, no.3, pp.463-468, 2016 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 23 Issue: 3
  • Publication Date: 2016
  • Doi Number: 10.1142/s1005386716000444
  • Journal Name: ALGEBRA COLLOQUIUM
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.463-468
  • Istanbul Technical University Affiliated: Yes

Abstract

In this paper, we study modules having only finitely many submodules over any ring which is not necessarily commutative. We try to understand how such a module decomposes as a direct sum. We justify that any module V having only finitely many submodules over any ring A is an extension of a cyclic A-module by a finite A-module. Under some assumptions on A, such as commutativity of A, we prove that an A-module V has finitely many submodules if and only if V can be written as a direct sum of a cyclic A-module having only finitely many A-submodules and a finite A-module.