Research Article Open Access

Modules in σ[M] with Chain Conditions on δM-small Submodules

Ali O. Alattass

Abstract

Problem statement: Let M be a right module over a ring R. In this article modules in σ[M] with chain conditions on δM- small submodules are studied. Approach: With the help of known results about M- singular, Artinian and Noetherian modules the techniques of the proofs of our main results use the properties of δM- small, δM- supplement and δM- semimaximal submodules. Results: Modules in σ[M] with chain conditions on δM- small are investigated, δM- semimaximal submodule is defined. Some Properties of δM- semimaximal submodules are proved. As application a new characterization of Artinian module in σ[M] is obtained in terms of δM- small submodules and δM- semimaximal submodules, as well as δM- small submodules and δM- supplement submodules. Conclusion/Recommendations: Our results certainly generalized several results obtained earlier.

Journal of Mathematics and Statistics
Volume 7 No. 2, 2011, 144-148

DOI: https://doi.org/10.3844/jmssp.2011.144.148

Submitted On: 6 March 2011 Published On: 18 May 2011

How to Cite: Alattass, A. O. (2011). Modules in σ[M] with Chain Conditions on δM-small Submodules. Journal of Mathematics and Statistics, 7(2), 144-148. https://doi.org/10.3844/jmssp.2011.144.148

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Keywords

  • Small submodules
  • supplement submodules
  • chain conditions
  • M-singular
  • supplemented module
  • finitely generated
  • uniform dimension
  • nonzero submodules
  • positive integer