The Radical Trace Property and Primary Ideals

Abstract

Let R be an integral domain and let M be an R-module. Then the trace ! Ž . 4 of M is the ideal generated by the set fm fHom M, R and mM . R For an ideal I of R, the trace is simply the product of I and I#1. We call an ideal J a trace ideal if J is the trace of some R-module. An elementary result which will be used freely throughout this paper is that if J is a trace #1 #1 Ž . $ % ideal, then JJ & J; i.e., J equals the ring J : J 6, Proposition 7.2 . Thus the trace ideals of R are precisely those ideals J for which J#1 Ž . Ž equals J : J . Such ideals are also referred to as being ‘‘strong’’; see, for $ % . example, 3 . If R is a valuation domain and M is an R-module, then the $ % trace of M is either R or a prime ideal of R 14, Proposition 2.1 . Extracting the conclusion of this result, Fontana et al. give the following Ž definition: A domain R is said to satisfy the trace property or to be a TP . domain if for each R-module M, the trace of M is equal to either R or a $ % Ž $ %. prime ideal of R 14, p. 169 see also 1, Theorem 2.8 . Theorem 3.5 of $ % 14 gives a characterization of Noetherian TP domains. Namely, for a Noetherian domain R, R is a TP domain if and only if R is onedimensional, has at most one non-invertible maximal ideal M, and, if such #1 Ž a maximal ideal exists, then M equals the integral closure of R or, #1 Ž . . $ % equivalently, M & M : M is a Dedekind domain . In Section 2 of 17 , Gabelli shows that by replacing ‘‘integral closure’’ with ‘‘complete integral closure,’’ the same list of conditions characterizes the class of Mori domains which satisfy the trace property. Recall that a Mori domain is an integral domain which satisfies the ascending chain condition on divisorial ideals.