On the saturations of submodules
Let \(R\subseteq S\) be a ring extension, and let \(A\) be an \(R\)-submodule of \(S\). The saturation of \(A\) (in \(S\)) by \(\tau\) is set \(A_{[\tau] }= \left\{x\in S : tx\in A \text{ for some } t\in \tau\right\}\), where \(\tau\) is a multiplicative subset of \(R\). We study properties of satu...
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| Datum: | 2018 |
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| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Lugansk National Taras Shevchenko University
2018
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| Schlagworte: | |
| Online Zugang: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/361 |
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| Назва журналу: | Algebra and Discrete Mathematics |
Institution
Algebra and Discrete Mathematics| Zusammenfassung: | Let \(R\subseteq S\) be a ring extension, and let \(A\) be an \(R\)-submodule of \(S\). The saturation of \(A\) (in \(S\)) by \(\tau\) is set \(A_{[\tau] }= \left\{x\in S : tx\in A \text{ for some } t\in \tau\right\}\), where \(\tau\) is a multiplicative subset of \(R\). We study properties of saturations of \(R\)-submodules of \(S\). We use this notion of saturation to characterize star operations \(\star\) on ring extensions \(R\subseteq S\) satisfying the relation \((A\cap B)^{\star} = A^{\star}\cap B^{\star}\) whenever \(A\) and \(B\) are two \(R\)-submodules of \(S\) such that \(AS= BS = S\). |
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