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...

The concept of a 2-Absorbing submodule is considered as an essential feature in the field module theory and has many generalizations. This articale discusses Extend Nearly Pseudo Quasi-2-Absorbing submodules their relationship to submodule, Nearly-2-Absorbing Pseudo-2-Absorbing rest other concepts previously studied. between them been studied, explaining that opposite not true under certain con...

This research aims to give the decompositions of a finitely generated module over some special ring, such as principal ideal domain and Dedekind domain. One main problems with theory is analyze objects module. was using literature study on modules topics from scientific journals, especially those related theory. And by selective cases we find pattern build conjecture or hypothesis, deductive pr...

Let be a module over commutative ring with identity. In this paper we intoduce the concept of Strongly Pseudo Nearly Semi-2-Absorbing submodule, where proper submodule an -module is said to if whenever , for implies that either or generalization 2_Absorbing semi 2-Absorbing and strong form (Nearly–2–Absorbing, Pseudo_2_Absorbing, Semi–2–Absorbing) submodules. Several properties characterization...

A module $M$ is called $emph{H}$-cofinitely supplemented if for every cofinite submodule $E$ (i.e. $M/E$ is finitely generated) of $M$ there exists a direct summand $D$ of $M$ such that $M = E + X$ holds if and only if $M = D + X$, for every submodule $X$ of $M$. In this paper we study factors, direct summands and direct sums of $emph{H}$-cofinitely supplemented modules. Let $M$ be an $emph{H}...

a module $m$ is called $emph{h}$-cofinitely supplemented if for every cofinite submodule $e$ (i.e. $m/e$ is finitely generated) of $m$ there exists a direct summand $d$ of $m$ such that $m = e + x$ holds if and only if $m = d + x$, for every submodule $x$ of $m$. in this paper we study factors, direct summands and direct sums of $emph{h}$-cofinitely supplemented modules. let $m$ be an $emph{h}$...