Pseudo-closure-operators (Q2639934)

Let M be a left module over an associative ring R with identity, and \(\Sigma\) (M) the set of all subsets of M properly containing the zero element of M. For \(S\in \Sigma (M)\), the pseudo-closure \(S^ c\) of S is defined as \(S^ c=\{0\}\cup \{m\in M\); Rm\(\cap S\neq 0\}\). Some elementary properties of the pseudo-closure operator \(S\mapsto S^ c\) on \(\Sigma\) (M), related to essential submodules, essentially closed submodules, the intersection property are given.