Trace properties and the rings $$R({\scriptstyle {\mathrm{X}}})$$ and $$R\langle {\scriptstyle {\mathrm{X}}}\rangle $$

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. $$I(R:I)R_P=PR_P$$ 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 $$R({\scriptstyle {\mathrm{X}}})$$ and the ring $$R\langle {\scriptstyle {\mathrm{X}}}\rangle $$ 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.