Star-reductions of ideals and Prufer v-multiplication domains

TL;DR

This paper extends the characterization of Prufer domains with the basic ideal property to Prufer v-multiplication domains using star operations, exploring their ideal reduction properties.

Contribution

It generalizes Hays' results by introducing w-basic properties for Prufer v-multiplication domains, linking ideal reductions with star operations.

Findings

Prufer v-multiplication domains characterized by w-basic ideal property

Relations among star-basic properties for specific star operations analyzed

Extension of classical ideal reduction concepts to broader domain classes

Abstract

Let R be a commutative ring and I an ideal of R. A sub-ideal J of 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) Prufer domains as domains with the finite basic ideal property (basic ideal property). We extend Hays' results to Prufer 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.