#### Document Type

Article

#### Department

Mathematics (HMC)

#### Publication Date

9-1994

#### 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 is a valuation domain for every prime ideal P of A. If M is a maximal *ℓ*-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*-ideals. If the latter is a positive integer, then A is said to have *finite* *rank*, and if 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), and X is called an *SV-space* if C(X) is an SV-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)) then if A is an SV-ring then it has finite rank. Showing that ihis 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.

#### Rights Information

© 1994 American Mathematical Society

#### Terms of Use & License Information

#### Recommended Citation

Henriksen, Melvin, Suzanne Larson, Jorge Martinez, and R. G. Woods. "Lattice-ordered algebras that are subdirect products of valuation domains." Transactions of the American Mathematical Society 345.1 (1994): 195-221.

## Comments

Previously linked to as: http://ccdl.libraries.claremont.edu/u?/irw,392.

Publisher pdf, posted with permission.

This article is also available at http://www.ams.org/journals/tran/1994-345-01/home.html.