Trace properties in rings with zero divisors

An integral domain D is said to have the radical trace property if I(D:I) is a radical ideal for each noninvertible nonzero ideal I of D. For a commutative ring R with nonzero identity, the dual of a dense ideal I is the set (R:I)={t∈Q(R)|tI⊆R} where Q(R) is the complete ring of quotients of R. In this article the radical trace property is extended to rings with nonzero zero divisors. Specifically, R has the radical trace property for regular ideals if I(R:I) is a radical ideal for each noninvertible regular ideal I; and R has the radical trace property for semiregular ideals if the same conclusion holds for noninvertible semiregular ideals (so to those ideals that contain finitely generated dense ideals). Alternately, R is said to be a RTP ring in the regular case and a Q0-RTP in the semiregular case. If R is an RTP ring and S is an overring of R (so between R and the total quotient ring T(R)) that is R-flat, then S is an RTP ring. A similar statement holds for Q0-RTP rings in the case S is R-flat and between R and the ring of finite fractions Q0(R).