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Reductions of ideals in Prüfer rings. (English) Zbl 1518.13002

Let \(R\) be a commutative ring and \(J\subseteq I\) ideals of \(R\). Then \(J\) is a reduction of \(I\) if \(JI^{n}=I^{n+1}\) for some positive integer \(n\), and \(I\) is basic if it has no proper reduction. Two well-known results, due to Hays, assert that an integral domain is Prüfer if and only if every finitely generated ideal is basic, and it is one-dimensional Prüfer if and only if every ideal is basic. The paper under review investigates reductions of ideals in the family of Prüfer rings (with zero-divisors), with the aim to recover and generalize Hays’ results. Two satisfactory results are obtained. The first result asserts that \(w. \dim(R)\leq 1\) if and only if R has the finite basic ideal property; and \(R\) is Prüfer if and only if \(R\) has the finite basic regular ideal property. The second result states that \(w. \dim(R)\leq 1\) and \(\dim(R) \leq 1\) if and only if R has the basic ideal property and if \(R\) is Prüfer with \(\dim(R) \leq 1\), then \(R\) has the basic regular ideal property. The converse is not true in general.

MSC:

13A15 Ideals and multiplicative ideal theory in commutative rings
13A18 Valuations and their generalizations for commutative rings
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
13F30 Valuation rings
13G05 Integral domains
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References:

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