The radical trace property. (English) Zbl 0641.13001
A domain R is said to satisfy TP (trace property) if for each ideal I of R, either \(II^{-1}=R\) or \(II^{-1}\in Spec(R)\). If \(II^{-1}=R\) or \(II^{-1}=rad(II^{-1})\) for each ideal I of R then R is said to satisfy RTP (radical trace property). If R is a Noetherian domain then R has RTP iff \(R_ P\) is a TP-domain for each \(P\in Spec(R)\) (proposition 2.1). This yields that integrally closed Noetherian RTP domains are Dedekind, RTP Krull domains are Dedekind, coherent integrally closed RTP domains are Prüfer. Further, the authors characterize the RTP Prüfer domains with acc for prime ideals. At the end of the paper some results concerning MTP domains \((II^{-1}=R\) or \(II^{-1}\in Max(R)\) for each ideal I of R) are presented.
Reviewer: L.Bican
MSC:
| 13A10 | Radical theory on commutative rings (MSC2000) |
| 13F05 | Dedekind, Prüfer, Krull and Mori rings and their generalizations |
| 13A15 | Ideals and multiplicative ideal theory in commutative rings |
| 13G05 | Integral domains |
| 13E05 | Commutative Noetherian rings and modules |
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