2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/142170This paper studies the notions of star and semistar operations over a polynomial ring. It aims at characterizing when every upper to zero in $R[X]$ is a $*$-maximal ideal and when a $*$-maximal ideal $Q$ of $R[X]$ is extended from $R$, that is, $Q=(Q\cap R)[X]$ with $Q\cap R\not =0$, for a given star operation of finite character $*$ on $R[X]$. We also answer negatively some questions raised by Anderson-Clarke by constructing a Prüfer domain $R$ for which the $v$-operation is not stable.9 pagesCommutative Algebra13A15; 13F20text