A characterization of Krull rings with zero divisors. (English) Zbl 0752.13014

Kang, Byung Gyun

J. Pure Appl. Algebra 72, No. 1, 33-38 (1991).

Let \(R\) be a commutative ring with identity, and let \(T(R)\) be the total quotient ring of \(R\). Then the notion of a Krull domain is generalized as follows: \(R\) is called a Krull ring if there exists a family \((V_ \alpha,P_ \alpha)\) of discrete rank one valuation pairs of \(T(R)\) with associated valuations \(v_ \alpha\) such that (1) \(R=\bigcap V_ \alpha\), (2) each \(P_ \alpha\) is a regular ideal of \(V_ \alpha\), and (3) for each regular element \(a\in T(R)\), \(v_ \alpha(a)\neq 0\) only for a finite number of \(\alpha\)’s. For a regular fractional ideal \(I\) one defines \(I_ t=\sum[R:[R:J]]\) where \(J\) runs over the finitely generated regular ideals contained in \(I\), and \(I\) is called \(t\)-invertible if \((II^{-1})_ t=R\). We quote the author’s main result:
Let \(R\neq T(R)\) and assume that \(R\) is a Marot ring, i.e. that every regular ideal of \(R\) is generated by regular elements; then \(R\) is a Krull ring if and only if every regular prime ideal contains a \(t\)- invertible prime ideal.

Reviewer: U.Vetter (Vechta)

MSC:

13F99	Arithmetic rings and other special commutative rings
13A15	Ideals and multiplicative ideal theory in commutative rings
13A18	Valuations and their generalizations for commutative rings

Keywords:

Krull ring; regular fractional ideal; \(t\)-invertible ideal; Marot ring; regular elements; valuation pairs

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