TY - JOUR

T1 - Trace Properties and Integral Domains, II

AU - Lucas, Thomas G.

AU - Mimouni, Abdeslam

PY - 2012/2

Y1 - 2012/2

N2 - 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.

AB - 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.

KW - LTP domain

KW - Radical trace property

KW - RTP domain

KW - Trace ideal

UR - http://www.scopus.com/inward/record.url?scp=84858148132&partnerID=8YFLogxK

U2 - https://doi.org/10.1080/00927872.2010.532525

DO - https://doi.org/10.1080/00927872.2010.532525

M3 - Journal article

SN - 0092-7872

VL - 40

SP - 497

EP - 513

JO - Communications in Algebra

JF - Communications in Algebra

IS - 2

ER -