t-Splitting sets in integral domains

Let D be an integral domain with quotient field K. A multiplicative subset S of D is a t-splitting set if for each 0≠d D, dD=(AB)t for some integral ideals A and B of D, where At∩sD=sAt for all s S and Bt∩S≠ ︀. A t-splitting set S of D is a t-lcm (resp., Krull) t-splitting set if sD∩dD is t-invertible (resp., sD is a t-product of height-one prime ideals of D) for all nonunitss S and 0≠d D. Let S be a t-splitting set of D, for some 0≠di D}, and for some . We show that S is a t-lcm (resp., Krull) t-splitting set if and only if is a PVMD (resp., Krull domain), if and only if every finite type integral v-ideal (resp., every integral ideal) of D intersecting S is t-invertible. We also show that D {0} is a t-splitting set in D[X] if and only if D is a UMT-domain and that every nonempty multiplicative subset of D[X] contained in G={f D[X] (Af)v=D} is a t-lcm t-complemented t-splitting set of D[X]. Using this, we give several Nagata-like theorems.