the submodules with the property of the title ( a submodule $n$ of an $r$-module $m$ is called strongly dense in $m$, denoted by $nleq_{sd}m$, if for any index set $i$, $prod _{i}nleq_{d}prod _{i}m$) are introduced and fully investigated. it is shown that for each submodule $n$ of $m$ there exists the smallest subset $d'subseteq m$ such that $n+d'$ is a strongly dense submodule of $m$...

‎let $r$ be a domain with quotiont field $k$‎, ‎and‎ ‎let $n$ be a submodule of an $r$-module $m$‎. ‎we say that $n$ is‎ ‎powerful (strongly primary) if $x,yin k$ and‎ ‎$xymsubseteq n$‎, ‎then $xin r$ or $yin r$ ($xmsubseteq n$‎ ‎or $y^nmsubseteq n$ for some $ngeq1$)‎. ‎we show that a submodule‎ ‎with either of these properties is comparable to every prime‎ ‎submodule of $m$‎, ‎also we show tha...

‎Let $R$ be a domain with quotiont field $K$‎, ‎and‎ ‎let $N$ be a submodule of an $R$-module $M$‎. ‎We say that $N$ is‎ ‎powerful (strongly primary) if $x,yin K$ and‎ ‎$xyMsubseteq N$‎, ‎then $xin R$ or $yin R$ ($xMsubseteq N$‎ ‎or $y^nMsubseteq N$ for some $ngeq1$)‎. ‎We show that a submodule‎ ‎with either of these properties is comparable to every prime‎ ‎submodule of $M$‎, ‎also we show tha...

Let be a module over commutative ring with identity. Before studying the concept of Strongly Pseudo Nearly Semi-2-Absorbing submodule, we need to mention ideal and basics that you study submodule. Also, introduce several characteristics submodule in classes multiplication modules other types modules. We also had no luck because is not ideal. it noted under conditions, which this faithful module...

We study the space of functions on a finite-dimensional vector space over a field of odd order as a module for a symplectic group. We construct a basis of this module with the following special properties. Each submodule generated by a single basis element under the symplectic group action is spanned as a vector space by a subset of the basis and has a unique maximal submodule. From these prope...

The submodules with the property of the title ( a submodule $N$ of an $R$-module $M$ is called strongly dense in $M$, denoted by $Nleq_{sd}M$, if for any index set $I$, $prod _{I}Nleq_{d}prod _{I}M$) are introduced and fully investigated. It is shown that for each submodule $N$ of $M$ there exists the smallest subset $D'subseteq M$ such that $N+D'$ is a strongly dense submodule of $M$ and $D'bi...

Let $G$ be a group with identity $e$, $R$ commutative $G$-graded ring unity $1$ and $M$ unital $R$-module. In this article, we introduce the concept of graded $1$-absorbing prime submodule. A proper $R$-submodule $N$ is said to if for all non-unit homogeneous elements $x, y$ element $m$ $xym\in N$, either $xy\in (N :_{R} M)$ or $m\in N$. We show that new generalization submodules at same time i...

A right module M over an associative ring with unity is a QTAG-module if every finitely generated submodule of any homomorphic image of M is a direct sum of uniserial modules. In this paper we find a suitable condition under which a special ω-elongation of a summable QTAG-module by a ( ω +k)-projective QTAG-module is also a summable QTAG-module.

All rings are commutative with 1 6= 0, and all modules are unital. The purpose of this paper is to investigate the concept of 2-absorbing primary submodules generalizing 2-absorbing primary ideals of rings. Let M be an R-module. A proper submodule N of an R-module M is called a 2-absorbing primary submodule of M if whenever a, b ∈ R and m ∈M and abm ∈ N , then am ∈M -rad(N) or bm ∈M -rad(N) or ...

Let M be a right module over a ring R. In this manuscript, we shall study on a special case of F-inverse split modules where F is a fully invariant submodule of M introduced in [12]. We say M is Z 2(M)-inverse split provided f^(-1)(Z2(M)) is a direct summand of M for each endomorphism f of M. We prove that M is Z2(M)-inverse split if and only if M is a direct...