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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.

MSC:

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

[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
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