## $$\star$$-reductions of ideals and Prüfer $$v$$-multiplication domains.(English)Zbl 1454.13006

Summary: 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. J. H. Hays [Trans. Am. Math. Soc. 177, 51–63 (1973; Zbl 0266.13001)] characterized (one-dimensional) Prüfer domains as domains with the finite basic ideal property (basic ideal property). We extend Hays’s results to Prüfer $$v$$-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 $$\star$$.

### MSC:

 13A15 Ideals and multiplicative ideal theory in commutative rings 13A18 Valuations and their generalizations for commutative rings 13C20 Class groups 13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations 13G05 Integral domains

Zbl 0266.13001
Full Text: