On Krull rings with zero divisors

Abstract

Let R be a commutative ring with identity and X-r(1) (R) the set of regular height one prime ideals of R. We will show that R is a Krull ring if and only if each regular prime ideal of R contains a t-invertible prime ideal, if and only if (1) R - boolean AND(P is an element of Xr1) ((R)) R-[P[ and it has finite character and (2) (R-[P], [P]R-[P]) is a rank one DVR for each P is an element of X-r(1) (R). It is also shown that {(R-[P], [P]R-[P]) P is an element of X-r(1) (R)} is the unique defining family for a Krull ring R.