This paper studies the notions of star and semistar operations over apolynomial 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 extendedfrom $R$, that is, $Q=(Q\cap R)[X]$ with $Q\cap R\not =0$, for a given staroperation of finite character $*$ on $R[X]$. We also answer negatively somequestions raised by Anderson-Clarke by constructing a Pr\"ufer domain $R$ forwhich the $v$-operation is not stable.
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机译：本文研究了多项式环上恒星和半星运算的概念。它旨在表征何时$ R [X] $的每个上到零都是$ * $的最大理想，以及$ R [X] $的$ * $的最大理想$ Q $从$ R $扩展而来，是，对于给定的$ R [X] $上有限字符$ * $的星号运算，$ Q =（Q \ cap R）[X] $且$ Q \ cap R \ not = 0 $。我们还通过构造$ v $操作不稳定的Pr'ufer域$ R $来否定安德森-克拉克提出的一些问题。
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