Abstract
31Aug 2016 Z- PRIMESUBMODULES. Nuhad Salim Al-Mothafar and Ali Talal Husain. Department of Mathematics, College of Science, Baghdad University, Baghdad, Iraq.
Highlights
Aproper submodule N of module M over ring R is said to be prime if rx ∈ N for r ∈ R and x ∈ M implies that either r ∈ N: M or x ∈ N where [N:M]= r ∈ R ∶ rM ⊆ N, [1].And M is called prime module if the zero submodule of M is a prime submodule of M, [2].In the last year many studies, researches are published about prime submodule by many people who care with the subject of commutative algebra and some of them are J..Dauns,R.L. .Mcsland, C.P.LU, P.F.Smith, M..E..Moore
In section two we define the Z-prime module and we show the relations between aZ-prime module and a faithful module, we see if R is integral domain R as R-module is Z-prime module, but the converse is not true in general
Corollary:LetN be a proper submodule of a multiplication R-module M, N is Z -prime submodule if and only if [N: M] is Z -prime ideal of R
Summary
Aproper submodule N of module M over ring R is said to be prime (or P-prime) if rx ∈ N for r ∈ R and x ∈ M implies that either r ∈ N: M or x ∈ N where [N:M]= r ∈ R ∶ rM ⊆ N , [1].And M is called prime module if the zero submodule of M is a prime submodule of M, [2].In the last year many studies, researches are published about prime submodule by many people who care with the subject of commutative algebra and some of them are J..Dauns ,R.L. .Mcsland, C.P.LU, P.F.Smith, M..E..Moore. We can find a faithful R-moule M but N is not a prime submodule of M, as the following example show: Let M = Z as Z-module and N = 4Z a submoduleof M. Proposition:Let M be a cyclic, faithful R-module and Nbe a submoduleofM, if N is aZ-prime submoduleof M, N is a prime submodule of M.
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