2009/07/24 by Naser Zamani, Zamani, Naser
Mathematics · #Advanced Topics in Algebra #Rings, Modules, and Algebras #math.AC #msc:13C05
paper · pdf · doi:10.48550/arxiv.0907.4349
8-pages, To apper in Glasgow Mathematical Journal
arxiv created 2009/07/24 · arxiv updated 2009/12/01
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 M is a prime submodule relative to ϕ or ϕ-prime submodule if a∈ R, x∈ M with ax∈ P∖ ϕ(P) implies that a∈(P:RM) or x∈ P. So if we take ϕ(N)=∅ for each N\inS(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 characterizations of ϕ-prime submodules will be given, and we show that under some assumptions prime submodules and ϕ1-prime submodules coincide.