$\star $-reductions of ideals and Prüfer $v$-multiplication domains

Abstract

Let $R$ be a commutative ring and $I$ an ideal of $R$. An ideal $J\subseteq I$ is a reduction of $I$ if $JI^{n}=I^{n+1}$ for some positive integer~$n$. The ring~$R$ has the (finite) basic ideal property if (finitely generated) ideals of $R$ do not have proper reductions. Hays characterized (one-dimensional) Pr\"ufer domains as domains with the finite basic ideal property (basic ideal property). We extend Hays's results to Pr\"ufer $v$-multiplication domains by replacing ``basic'' with ``$w$-basic,'' where $w$ is a particular star operation. We also investigate relations among $\star $-basic properties for certain star operations $\star $.