SOME PROPERTIES OF STAR OPERATIONS ON RING EXTENSIONS

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