On Krull rings with zero divisors

Let R be a commutative ring with identity and Xr1(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=∩P∈Xr1(R)R[P] and it has finite character and (2) (R[P],[P]R[P]) is a rank one DVR for each P∈Xr1(R). It is also shown that {(R[P],[P]R[P])|P∈Xr1(R)} is the unique defining family for a Krull ring R.