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Modules with finitely many submodules
, Article Algebra Colloquium ; Volume 23, Issue 3 , 2016 , Pages 463-468 ; 10053867 (ISSN) ; Khalashi Ghezelahmad, S ; Yaraneri, E ; Sharif University of Technology
World Scientific Publishing Co. Pte Ltd
2016
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