Pseudo-closure-operators. (English) Zbl 0719.16002

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.


16D10 General module theory in associative algebras
06A15 Galois correspondences, closure operators (in relation to ordered sets)
Full Text: Numdam EuDML


[1] Y. Miyashita , On quasi-injective modules , Journ. of the Fac. of Sc. Hokkado University , Ser. 1 , Math. , 18 ( 1965 ), pp. 158 - 187 . MR 171817 | Zbl 0199.07801 · Zbl 0199.07801
[2] B. Stenström , Rings of Quotients , Springer , 1975 . MR 389953 | Zbl 0296.16001 · Zbl 0296.16001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.