‎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 commutative ring with identity. A proper submodule $N$ of an $R$-module $M$ is an n-submodule if $rmin N~(rin R, min M)$ with $rnotinsqrt{Ann_R(M)}$, then $min N$. A number of results concerning n-submodules are given. For example, we give other characterizations of n-submodules. Also various properties of n-submodules are considered.

a celebrated result of i. schur asserts that the derived subgroup of a group is finite provided the group modulo its center is finite, a result that has been the source of many investigations within the theory of groups. in this paper we exhibit a similar result to schur's theorem for vector spaces, acted upon by certain groups. the proof of this analogous result depends on the characteristic o...

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 $m$...

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 ...

In this paper we investigate decompositions of submodules in modules over a Proufer domain into intersections of quasi-primary and classical quasi-primary submodules. In particular, existence and uniqueness of quasi-primary decompositions in modules over a Proufer domain of finite character are proved. Proufer domain; primary submodule; quasi-primary submodule; classical quasi-primary; decompo...

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...