Lattice-ordered algebras that are subdirect products of valuation domains

Abstract

An f-ring (i.e., a lattice-ordered ring that is a subdirect product of totally ordered rings) A is called an SV-ring if A / P A/P is a valuation domain for every prime ideal P of A . If M is a maximal ℓ \ell -ideal of A, then the rank of A at M is the number of minimal prime ideals of A contained in M , rank of A is the sup of the ranks of A at each of its maximal ℓ \ell -ideals. If the latter is a positive integer, then A is said to have finite rank, and if A = C ( X ) A = C(X) is the ring of all real-valued continuous functions on a Tychonoff space, the rank of X is defined to be the rank of the f-ring C ( X ) C(X) , and X is called an SV-space if C ( X ) C(X) is an ST-ring. X has finite rank k iff k is the maximal number of pairwise disjoint cozero sets with a point common to all of their closures. In general f-rings these two concepts are unrelated, but if A is uniformly complete (in particular, if A = C ( X ) A = C(X) ) then if A is an SV-ring then it has finite rank. Showing that this latter holds makes use of the theory of finite-valued lattice-ordered (abelian) groups. These two kinds of rings are investigated with an emphasis on the uniformly complete case. Fairly powerful machinery seems to have to be used, and even then, we do not know if there is a compact space X of finite rank that fails to be an SV-space.