Trace properties in integral domains: a survey

Abstract

An integral domain R is a TP domain (or satisfies the trace property) if the trace of each R-module is equal to either R or a prime ideal of R. Equivalently, every trace ideal is a prime ideal of R, that is, for every non-zero non-invertible (fractional) ideal I of R,\ I(R:I) is a prime ideal of R.The notion of radical trace property relaxed the requirement that each trace ideal be a prime ideal to require only that each trace ideal is a radical ideal. Equivalently, a domain R is an RTP domain (or has theradical trace property) if I(R:I) is a radical ideal for each nonzero non-invertible ideal I. Two other notions related to trace property are the notion of trace property for primary ideals and L-trace property. A domain is a TPP (resp. LTP) domain if Q(R:Q)=R or Q(R:Q) is a prime ideal of R for every primary ideal Q of R\ (resp. I(R:I)R_P=PR_P for each minimal prime P of I(R:I)\,). Clearly each TP domain is an RTP domain, but not conversely. Also each RTP domain is a TPP domain and each TPP domain is an LTP domain, but whether the three notions RTP, TPP and LTP are equivalent is open except in certain special cases. This survey paper tracks some old/recent works investigating these notionsin different contexts of integral domains such as integrally closed domains (namely valuation and Prüfer domains), Noetherian and Mori domains, pseudo-valuation domains and pullbacks, and Nagata and Serre’s conjecture rings.

Keywords:  trace ideal, radical trace property, RTP domain, LTP domain.

MSC numbers: Primary 13A15; Secondary 13F05, 13G05.

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