## 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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