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φ-PRIME SUBMODULES

Published online by Cambridge University Press:  25 November 2009

NASER ZAMANI
NASER ZAMANI*
Affiliation:
Faculty of Science, University of Mohaghegh Ardabili, P.O. Box 179, Ardabil, Iran e-mail: naserzaka@yahoo.com, zamanin@uma.ac.ir
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Abstract

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Let R be a commutative ring with non-zero identity and M be a unitary R-module. Let (M) be the set of all submodules of M, and φ: (M) → (M) ∪ {∅} be a function. We say that a proper submodule P of M is a prime submodule relative to φ or φ-prime submodule if a ∈ R and x ∈ M, with ax ∈ P ∖ φ(P) implies that a ∈(P :RM) or x ∈ P. So if we take φ(N) = ∅ for each N ∈ (M), then a φ-prime submodule is exactly a prime submodule. Also if we consider φ(N) = {0} for each submodule N of M, then in this case a φ-prime submodule will be called a weak prime submodule. Some of the properties of this concept will be investigated. Some characterisations of φ-prime submodules will be given, and we show that under some assumptions prime submodules and φ1-prime submodules coincide.

Keywords

13C05
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 52 , Issue 2 , May 2010 , pp. 253 - 259
Copyright
Copyright © Glasgow Mathematical Journal Trust 2009

References

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