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.