Rees valuations and asymptotic primes of rational powers in Noetherian rings and lattices

We extend a theorem of D. Rees on the existence of Rees valuations of an ideal A of a Noetherian ring to Noetherian multiplicative lattices L. This result also extends a result of D.P. Brithinee. We then apply this to projective equivalence and asymptotic primes of rational powers of A. In particular, it is shown that if L is a Noetherian multiplicative lattice, A∈L, {P1,…,Pr} is the set of centers of the Rees valuations v1,…,vr of A and e is the least common multiple of the Rees numbers e1(A),…,er(A) of A, then Ass(L/An/e)⊆{P1,…,Pr}, where . Further, if A⩽̸q for each minimal prime q∈L, then Ass(L/An/e)⊆Ass(L/An/e+k/e) for each n∈N, where k/e is in a certain additive subsemigroup of Q+ which is naturally associated to the set of members of L which are projectively equivalent to A. These latter results are new even in the case of rings and extend results of L.J. Ratliff who gave them for rings in the case that the n/e and k/e are integers.