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- Title
φ-PRIME SUBMODULES.
- Authors
Zamani, Naser
- Abstract
Let R be a commutative ring with non-zero identity and M be a unitary R-module. Let S(M) be the set of all submodules of M, and φ : S(M) → S(M) ? {∅} be a function. We say that a proper submodule P of Mis 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 :R M) or x ∈ P. So if we take φ(N) = ∅ for each N ∈ S(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.
- Subjects
COMMUTATIVE rings; RING theory; INTEGRAL closure; COMMUTATIVE algebra; NUMERICAL analysis; MATHEMATICAL analysis
- Publication
Glasgow Mathematical Journal, 2010, Vol 52, Issue 2, p253
- ISSN
0017-0895
- Publication type
Article
- DOI
10.1017/S0017089509990310