Let G be a group with identity e. R G-graded commutative ring and M graded R-module. We introduce the concept of Ie-prime submodule as generalization prime for I =?g?G Ig fixed ideal R. give number results concerning this class submodules their homogeneous components. A proper N is said to if whenever rg ? h(R) mh h(M) rgmh IeN, then either (N :R M) or N.

Let $R$ be a commutative ring with identity and $M$ be a unitary $R$-module. Suppose that $phi:S(M)rightarrow S(M)cup lbraceemptysetrbrace$ be a function where $S(M)$ is the set of all submodules of $M$. A proper submodule $N$ of $M$ is called an $(n-1, n)$-$phi$-classical prime submodule, if whenever $r_{1},ldots,r_{n-1}in R$ and $min M$ with $r_{1}ldots r_{n-1}min Nsetminusphi(N)$, then $r_{1...

Let $G$ be a group, $R$ $G$-graded commutative ring with identity, $M$ graded $R$-module and $S\subseteq h(R)$ multiplicatively closed subset of $R$. In this paper, new concept $S$-primary submodules is introduced as generalization Primary well $S$-prime $M$. Also, some properties class are investigated.

Let $G$ be a group, $R$ $G$-graded commutative ring with identity, $M$ graded $R$-module and $S\subseteq h(R)$ multiplicatively closed subset of $R$. In this paper, new concept $S$-primary submodules is introduced as generalization Primary well $S$-prime $M$. Also, some properties class are investigated.

In this paper we introduce the notions of uniformly quasi-primary ideals and uniformly classical quasi-primary submodules that generalize the concepts of uniformly primary ideals and uniformly classical primary submodules; respectively. Several characterizations of classical quasi-primary and uniformly classical quasi-primary submodules are given. Then we investigate for a ring $R$, when any fi...

let $g$ be a group with identity $e$. let $r$ be a $g$-graded commutative ring with a non-zero identity and $m$ be a graded $r$-module. in this article, we introduce the concept of graded almost semiprime submodules. also, we investigate some basic properties of graded almost semiprime and graded weakly semiprime submodules and give some characterizations of them.

Let $G$ be a group with identity $e$. $R$ $G$-graded commutative ring and $M$ graded $R$-module. In this paper, we introduce the concept of classical strongly 2-absorbing second submodules modules over rings. A number results concerning these classes their homogeneous components are given.