Splitting sets in integral domains

Let D be an integral domain. A saturated multiplicatively closed subset S o f D is a splitting set if each nonzero d is an element of D may be written as d = sa where s is an element of S and s'D boolean AND aD = s'aD for all s' is an element of S. We show that if S is a splitting set in D, then SU(D -N) is a splitting set in D-N, N a multiplicatively closed subset of D, and that S subset of or equal to D is a splitting set in D[X] double left righ t arrow S is an lcm splitting set of D, i.e., S is a splitting set of D wit h the further property that sD boolean AND dD is principal for all s is an element of S and d is an element of D. Several new characterizations and ap plications of splitting sets are given.