SOME PROPERTIES OF STAR OPERATIONS ON RING EXTENSIONS

Abstract

Let $\star$ be a star operation on a ring extension $R\subseteq S$. A ring extension $R\subseteq S$ is called Pr\"ufer $star$-multiplication extension (P$\star$ME) if $(R_{[\m]}, \m _{[\m]})$ is a Manis pair in $S$ for every $\star$-maximal ideal $\m$ of $R$. We establish some results on star operations, and we study P$\star$ME in pullback diagrams of type $\square$. We show that, for a maximal ideal $\m$ of $R$, the extension $R_{[\m]} \subseteq S$ is Manis if and only if $R[X]_{[\m R[X]]} \subseteq S[X]$ is a Manis extension.