Modules with Finitely Many Submodules

Author:

Akbari S.12,Ghezelahmad S. Khalashi3,Yaraneri E.4

Affiliation:

1. Department of Mathematical Sciences, Sharif University of Technology, P.O. Box 11155-9415, Tehran, Iran

2. School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran

3. School of Mathematics, Iran University of Science and Technology, Tehran, Iran

4. Department of Mathematics, Istanbul Technical University, Istanbul, Turkey

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.

Publisher

World Scientific Pub Co Pte Lt

Subject

Applied Mathematics,Algebra and Number Theory

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