Author/Authors :
Ebrahimpour، M. نويسنده Shahid Bahonar University of Kerman , , Nekooei، R. نويسنده Shahid Bahonar University of Kerman ,
Abstract :
Let $R$ be a commutative ring with identity and $M$ be a unitary
$R$-module. Let $\phi:S(M)\rightarrow S(M)\cup\{\emptyset\}$ be a
function, where $S(M)$ is the set of submodules of $M$. Suppose
$n\geq 2$ is a positive integer. A proper submodule $P$ of $M$ is
called $(n-1,n)-\phi$-prime, if whenever $a_1,\dots,a_{n-1}\in R$
and $x\in M$ and $a_1\dots a_{n-1}x\in P\backslash\phi(P)$, then
there exists $i\in\{1,\dots,n-1\}$ such that $a_1\dots
a_{i-1}a_{i+1}\dots a_{n-1}x\in P$ or $a_1\dots a_{n-1}\in(P:M)$.
In this paper we study $(n-1,n)-\phi$-prime submodules $(n\geq
2)$. A number of results concerning $(n-1,n)-\phi$-prime
submodules are given. Modules with the property that for some
$\phi$, every proper submodule is $(n-1,n)-\phi$-prime, are
characterized and we show that under some assumptions
$(n-1,n)$-prime submodules and $(n-1,n)-\phi_m$-prime submodules
coincide ($n,m\geq 2$).
