Abstract
This article 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 ∩ R)[X] with Q ∩ R ≠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.
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