On strongly dense submodules‎

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 $D'bigcap N=0$. We also introduce a class of modules in which the two concepts of strong essentiality and strong density coincide. It is also shown that for any module $M$, dense submodules in $M$ are strongly dense if and only if $Mleq_{sd} tilde{E}(M)$, where $tilde{E}(M)$ is the rational hull of $M$. It is proved that $R$ has no strongly dense left ideal if and only if no nonzero-element of every cyclic $R$-module $M$ has a strongly dense annihilator in $R$. Finally, some appropriate properties and new concepts related to strong density are defined and studied.