On prime divisors of large powers of elements in Noether lattices

Abstract

In [4], R. P. Dilworth introduced the concept of a Noether lattice as an abstraction of the lattice of ideals of a Noetherian ring and he showed that many important properties of Noetherian rings, such as the Noether decomposition theorems, also hold for Noether lattices. It was later shown, in [1], that every Noether lattice is not the lattice of ideals of any Noetherian ring, yet many studies have successfully been undertaken to relate other concepts between Noetherian rings and Noether lattices as had been begun by Dilworth. (See [3], [5], and [6].) In this paper we undertake such a study and show that some results of M. Brodmann in [2] and L. Ratliff in [7] concerning prime divisors of large powers of a fixed element of a commutative Noetherian ring may be generalized and extended to the setting of a Noether lattice. It is shown (Theorem 2.8) that if A is an element of a Noether lattice then all large powers of A have the same prime divisors and (Corollary 3.8) included among this fixed set of primes are those primes that are prime divisors of the integral closure of Ak for some k≧l. We note that the ring proof of this latter result does not generalize directly since it uses the notion of transcendence degree which to our knowledge has no analogue in multiplicative lattices.