  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.
