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...

Let $R$ be a commutative ring and let $M$ be an $R$-module. We define the small intersection graph $G(M)$ of $M$ with all non-small proper submodules of $M$ as vertices and two distinct vertices $N, K$ are adjacent if and only if $Ncap K$ is a non-small submodule of $M$. In this article, we investigate the interplay between the graph-theoretic properties of $G(M)$ and algebraic properties of $M...

The notion of intuitinitic (T, S)-fuzzy submodule in Module is introduced, and related properties are investigated. Mathematics Subject Classification: 08A72, 13C99

A module MR is called right principally quasi-Baer (or simply right p.q.-Baer) if the right annihilator of a principal submodule of R is generated by an idempotent. Let R be a ring. Let α be an endomorphism of R and MR be a α-compatible module and T = R[[x;α]]. It is shown that M [[x]]T is right p.q.-Baer if and only if MR is right p.q.-Baer and the right annihilator of any countably-generated ...

Let $R$ be a commutative ring and $M$ an $R$-module. A submodule $N$ of is called d-submodule $($resp., fd-submodule$)$ if $\ann_R(m)\subseteq \ann_R(m')$ $\ann_R(F)\subseteq \ann_R(m'))$ for some $m\in N$ finite subset $F\subseteq N)$ $m'\in M$ implies that N.$ Many examples, characterizations, properties these submodules are given. Moreover, we use them to characterize modules satisfying Prop...

Abstract In previous research, a concept was presented f –small submodules A submodule L of an R-module W is named if L+X=W and W/X are small-singular, then X=W. this work, we present new concept, namely –hollow modules, -hollow every in -small submodule. This expansion Hollow modules. Some characteristics given to their relationship few the concepts comparability between them examined.

Mat r x ( F q ) o f a l l r b y r m a t r i c e s o v e r F q , b y C t h e g r o u p G L t i ( F q ) o f t h e u n i t s of M, and by S the special linear subgroup SL , (F q ) o f C . F o r a n a r b i t r a r y field F containing F q , l e t U s t a n d f o r t h e ( c o m m u t a t i v e ) p o l y n o m i a l a l g e b r a F[xl, , x r ] , a n d c o n s i d e r U g r a d e d a s u s u a l : U...

The purpose of this paper is to introduce new concepts in a module  over ring , the first one called e*- small essential submodule, which generalization second concept called e*-radical submodule radical and last e* - hollow module. We will prove some properties all these concepts.