Trace Properties and Integral Domains, II

Thomas G. Lucas, Abdeslam Mimouni

Research output: Contribution to journalJournal articlepeer-review

2 Scopus citations

Abstract

An integral domain R has the radical trace property, or is an RTP domain, if I(R: I) is a radical ideal for each nonzero noninvertible ideal I. A related property is LTP: I(R: I)R P = PR P for each minimal prime P of I(R: I). It is clear that each RTP domain is an LTP domain, but whether the two are equivalent is open except in certain special cases. The ideal I(R: I) is the trace of I and I is a trace ideal if I(R: I) = I. A new characterization for RTP domains is established for Noetherian domains, Mori domains, and Prüfer domains. In these special cases, R is an RTP domain if and only if IB(R: IB) = I for each trace ideal I and each ideal B of (R: I) that contains I (here (R: I) is a ring as I is a trace ideal). In the Prüfer case, having IJ(R: IJ) = I for each trace ideal I and each ideal J of R containing I is enough to have the radical trace property. On the other hand, an example is given of a one-dimensional local Noetherian domain R that is not an RTP domain but does satisfy IJ(R: IJ) = I for each trace ideal I and each ideal J of R that contains I.

Original languageEnglish
Pages (from-to)497-513
Number of pages17
JournalCommunications in Algebra
Volume40
Issue number2
DOIs
StatePublished - Feb 2012

Keywords

  • LTP domain
  • Radical trace property
  • RTP domain
  • Trace ideal

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