Unique factorization and small regular local noether lattices

Noether lattices were introduced by Dilworth [5]. Simply stated, a Noether lattice is a complete, modular, commutative multiplicative lattice satisfying the ascending chain condition in which every element is the join of principal elements. The resultant setting not only admits natural formulations of the fundamental definitions and results of classical ideal theory, but is simultaneously rich enough to yield, for example, the Noether decomposition theorems, the Intersection Theorem, and the Principal Ideal Theorem [Sj. The key to the richness of the setting lies, of course, in the definition of a principal element. An element E is said to be principal if it satisfies the two identities BE A A = (B A (A : E))E and (B v AE) : E = (B : E) v A, for all A and B. These identities are easily verified for the principal ideals of a commutative ring. In [4], Bogart obtained the surprising result that a regular local Noether lattice with exactly n < 3 rank 1 primes is isomorphic to RL, , the Noether sublattice of k[x, ,..., x,] (k a field) consisting of all ideals which are generated by monomials in x1 , xs ,..., x, . He conjectured the validity of the result for all n, but noted that, due to the heavy dependence of the case n = 3 on the observation that in a regular local Noether lattice of dimension 2, all rank 1 primes are principal, the general case might be closely tied to a generalization of the Auslander-Buchsbaum theorem that regular local rings are UFDs. In [2], Anderson conjectured that any local Noether lattice with exactly n rank 1 primes which is a n-domain (i.e., every principal element is a product of primes, or, equivalently in this case, every rank 1 prime is principal) is isomorphic to RL, , and therefore regular. In this paper we establish the validity of both conjectures. While our results do not yield a generalization of the Auslander-Buchsbaum theorem as hoped, they do at least show that any counterexample to the generalization will involve an infinite number of primes.