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Multiplicative closure operations on ring extensions
Journal of Pure and Applied Algebra ( IF 0.7 ) Pub Date : 2021-04-01 , DOI: 10.1016/j.jpaa.2020.106555
Dario Spirito

Let $A\subseteq B$ be a ring extension and $\mathcal{G}$ be a set of $A$-submodules of $B$. We introduce a class of closure operations on $\mathcal{G}$ (which we call \emph{multiplicative operations on $(A,B,\mathcal{G})$}) that generalizes the classes of star, semistar and semiprime operations. We study how the set $\mathrm{Mult}(A,B,\mathcal{G})$ of these closure operations vary when $A$, $B$ or $\mathcal{G}$ vary, and how $\mathrm{Mult}(A,B,\mathcal{G})$ behave under ring homomorphisms. As an application, we show how to reduce the study of star operations on analytically unramified one-dimensional Noetherian domains to the study of closures on finite extensions of Artinian rings.

中文翻译:

环扩展上的乘法闭包运算

令 $A\subseteq B$ 是环扩展,$\mathcal{G}$ 是 $B$ 的一组 $A$-submodules。我们在 $\mathcal{G}$ 上引入了一类闭包运算(我们称之为 \emph{$(A,B,\mathcal{G})$} 上的乘法运算),它概括了星形、半星形和半质数的类操作。我们研究了当 $A$、$B$ 或 $\mathcal{G}$ 变化时,这些闭包运算的集合 $\mathrm{Mult}(A,B,\mathcal{G})$ 如何变化,以及 $\ mathrm{Mult}(A,B,\mathcal{G})$ 在环同态下表现。作为一个应用,我们展示了如何将分析无分枝的一维诺特域上的星运算研究减少到对阿蒂尼环有限扩展的闭包的研究。
更新日期:2021-04-01
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