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 N≤sdM, if for any index set I, ∏IN≤d∏IM) are introduced and fully investigated. It is shown that for each submodule N of M there exists the smallest subset D′⊆M such that N+D′ is a strongly dense submodule of M and D′⋂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 M≤sdE~(M), where 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