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

Houston, E.; Kabbaj, S.; Mimouni, A.

J. Commut. Algebra 9, No. 4, 491-505 (2017).

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

Keywords:

star operation; P\(v\)MD; Prüfer domain; reduction of an ideal; \(\star\)-reduction of an ideal; basic ideal; \(\star\)-basic ideal; basic ideal property; \(\star\)-basic ideal property

Citations:

Zbl 0266.13001