Throughout this paper, R will denote a commutative ring with identity and M is a unitary R- module and Z will denote the ring of integers. We introduce the graph Ω(M) of module M with the set of vertices contain all nontrivial non-essential submodules of M. We investigate the interplay between graph-theoretic properties of Ω(M) and algebraic properties of M. Also, we assign the values of natura...

By considering the notion of multiplication modules over a commutative ring with identity, first we introduce the notion product of two submodules of such modules. Then we use this notion to characterize the prime submodules of a multiplication module. Finally, we state and prove a version of Nakayama lemma for multiplication modules and find some related basic results. 1. Introduction. Let R b...

In this paper we introduce a new class of modules which is closely related to the class of Noetherian modules. Let $R$ be a commutative ring with identity and let $M$ be an $R$-module such that $Nil(M)$ is a divided prime submodule of $M$. $M$ is called a Nonnil-Noetherian $R$-module if every nonnil submodule of $M$ is finitely generated. We prove that many of the properties of Noetherian modul...

We call a module M , quasi-catenary if for each pair of quasi-prime submodules K and L of M with K L all saturated chains of quasi-prime submodules of M from K to L have a common finite length. We show that any homomorphic image of a quasi-catenary module is quasi-catenary. We prove that if M   is a module with following properties: (i) Every quasi-prime submodule of M has finite quasi-height;...

Let G be a group with identity e. Let R be a G-graded commutative ring and let M be a graded R-module. The graded classical prime spectrum Cl.Specg(M) is defined to be the set of all graded classical prime submodule of M. The Zariski topology on Cl.Specg(M); denoted by ϱg. In this paper we establish necessary and sufficient conditions for Cl.Specg(M) with the Zariski topology to be a Noetherian...

let $r$ be an arbitrary ring and $t$ be a submodule of an $r$-module $m$. a submodule $n$ of $m$ is called $t$-small in $m$ provided for each submodule $x$ of $m$, $tsubseteq x+n$ implies that $tsubseteq x$. we study this mentioned notion which is a generalization of the small submodules and we obtain some related results.

One of the sciences used in digital security systems is cryptography. Cryptography closely related to integer system, especially prime numbers. Prime numbers themselves have been abstracted a lot. form abstraction ideal. Previous studies proven that an Ideal said be ideal on if and only I constructed by element. Other also shown how develops. them research result Dauns, where developed submodul...

Let $G$ be an abelian group with identity $e$. $R$ a $G$-graded commutative ring identity, $M$ graded $R$-module and $S\subseteq h(R)$ multiplicatively closed subset of $R$. In this paper, we introduce the concept $S$-prime submodules modules over rings. We investigate some properties class their homogeneous components. $N$ submodule such that $(N:_{R}M)\cap S=\emptyset $. say is \textit{a }$S$...

In this paper, we introduce the concept of a quasi-radical semi prime submodule. Throughout work, assume that is commutative ring with identity and left unitary R- module. A proper submodule called (for short Q-rad-semiprime), if for , ,and then . Where intersection all submodules