Trace properties in rings with zero divisors

Abstract

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 ) | t I ⊆ 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 Q 0 -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 Q 0 -RTP rings in the case S is R-flat and between R and the ring of finite fractions Q 0 ( R ) .