Trace properties and the rings $$R({\scriptstyle {\mathrm{X}}})$$ R ( X ) and $$R\langle {\scriptstyle {\mathrm{X}}}
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Trace properties and the rings R(X) and R⟨X⟩ Thomas G. Lucas1 · Abdeslam Mimouni2 Received: 17 September 2019 / Accepted: 8 February 2020 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract An integral domain R is an RTP domain (or has the radical trace property) (resp. an LTP domain), if I(R : I) is a radical ideal for each nonzero noninvertible ideal I (resp. I(R ∶ I)RP = PRP for each minimal prime P of I(R : I)). Clearly each RTP domain is an LTP domain, but whether the two are equivalent is open except in certain special cases. In this paper, we study when the Nagata ring R(X) and the ring R⟨X⟩ are LTP (resp. RTP) domains in different contexts of integral domains such as integrally closed domains, Noetherian and Mori domains, pseudo-valuation domains and more. We also study the descent of these notions from particular overrings of R to R itself. Keywords Trace ideal · Radical trace property · RTP domain · LTP domain Mathematics Subject Classification Primary 13A15 · Secondary 13F05 · 13G05
1 Introduction Throughout this article, R denotes an integral domain with quotient field K and integral closure R′ . For a nonzero fractional ideal I of R, (R ∶ I) = {x ∈ K|xI ⊆ R} is the dual of I and Iv = (R ∶ (R ∶ I)) is the divisorial closure of I (both “with respect to R”). The trace of an R-module B is the ideal of R generated by the set {𝜑(b) ∣ b ∈ B, 𝜑 ∈ HomR (B, R)} (see, for example, [13]). We say that an ideal I of R is a trace ideal if it is the trace of some R-module. In such a case, I will in fact be its own trace [6, Proposition 7.2], equivalently (R ∶ I) = (I ∶ I) . Thus we may restrict our study of “trace properties” to the noninvertible ideals of R.
* Abdeslam Mimouni [email protected] Thomas G. Lucas [email protected] 1
Department of Mathematics and Statistics, University of North Carolina Charlotte, Charlotte, NC 28223, USA
2
Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
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T. G. Lucas, A. Mimouni
In 1987, Anderson, Huckaba and Papick proved that the trace of each noninvertible ideal of a valuation domain is prime [1]. Soon after, Fontana, Huckaba and Papick introduced the trace property and TP domains [13]. They defined a TP domain to be an integral domain for which the trace of each noninvertible ideal is prime; alternately, one may say that the domain has the trace property. The notion of an RTP domain as an integral domain for which the trace of each noninvertible ideal is a radical ideal was introduced by Heinzer and Papick [19]; these are the domains with the radical trace property. Two types of domains that are related to RTP domains are TPP domains and LTP domains, introduced in [25] and [22], respectively. A TPP domain is one for which the trace of each noninvertible primary ideal is prime (in fact is its radical [25, Corollary 8]), and an LTP domain is a domain R for which IRP = PRP for each minimal prime P of a
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