Domains satisfying the trace property. (English) Zbl 0612.13010
Let R be a commutative integral domain with identity. R is said to satisfy the trace property if for each R-module M, the trace ideal of M is either equal to R or is a prime ideal. It is shown that R satisfies the trace property if and only if for each nonzero ideal I of R, either \(II^{-1}=R\) or is a prime ideal. It was shown by the reviewer and J. A. Huckaba and I. J. Papick [Houston J. Math. 13, 13-17 (1987)], that this last condition holds for a valuation domain. The purpose of this paper is to characterize certain classes of domains satisfying the trace property. It is shown that a Noetherian domain R satisfies the trace property if and only if R is Dedekind or \(\dim (R)=1\), R has a unique non-invertible maximal ideal M and \(M^{-1}\) is equal to the integral closure of R. The question of when a Prüfer domain satisfies the trace property is also considered.
Reviewer: D.D.Anderson
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