A study of linked star operations
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Let $R\subseteq L \subseteq S$ be ring extensions. Two star operations $\star_{1} \in \st (R, S)$, $\star_{2} \in \st (L, S)$ are said to be linked if whenever $A^{\star_{1}}=R^{\star_{1}}$ for some finitely generated $S$-regular $R$-submodule $A$ of $S$, then $(AL)^{\star_{2}} =L^{\star_{2}}$. We study properties of linked star operations; especially when $\star_{1}$ and $\star_{2}$ are strict star operations. We introduce the notion of Pr\"ufer star multiplication extension (P$\star$ME) and we show that under appropriate conditions, if the extension $R\subseteq S$ is P$\star _{1}$ME and $\star_{1}$ is linked to $\star_{2}$, then $L\subseteq S$ is P$\star _{2}$ME.
Abstract
Let $R\subseteq L \subseteq S$ be ring extensions. Two star operations $\star_{1} \in \st (R, S)$, $\star_{2} \in \st (L, S)$ are said to be linked if whenever $A^{\star_{1}}=R^{\star_{1}}$ for some finitely generated $S$-regular $R$-submodule $A$ of $S$, then $(AL)^{\star_{2}} =L^{\star_{2}}$. We study properties of linked star operations; especially when $\star_{1}$ and $\star_{2}$ are strict star operations. We introduce the notion of Pr\"ufer star multiplication extension (P$\star$ME) and we show that under appropriate conditions, if the extension $R\subseteq S$ is P$\star _{1}$ME and $\star_{1}$ is linked to $\star_{2}$, then $L\subseteq S$ is P$\star _{2}$ME.
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