Lattice-ordered algebras that are subdirect products of valuation domains

Abstract

Abstract. An /-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 . lî M is a maximal I -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 ¿-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 /-ring C(X), and X is called an SV-space if 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 /-rings these two concepts are unrelated, but if A is uniformly complete (in particular, if A = C(X)) then if A is an SF-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 SK-space.

Gabriel filters in real closed rings

“…For rings of continuous functions an answer has been known for a long time: Given a completely regular space X, let C(X) be the ring of continuous functions into R. Then every ideal of C(X) is an l-ideal if and only if X is an F -space, if and only if every prime ideal of C(X) contains a unique minimal prime ideal ( [15], Theorem 14.25). If this is the case then C(X)/p is a convex subring of the real closed field qf (C(X)/p) (which follows from [15], Theorem 4.7 and Theorem 14.24), i.e., C(X) is an SV -ring in the terminology of [22]. Proof.…”

“…So it remains prove that it is a multiplicative filter. by the localizations of rings C(X) of continuous functions from an SV -space X into the real numbers ( [22], Theorem 4.1).…”

Images and Open Subspaces of SV Spaces

“…It was stated, but not shown, in Henriksen et al (1994) that Y has a point of infinite rank. To show this, we first show that for any h ∈ C * L 2 × L 1 − 2 1…”

“…We let the rank of the f -ring A be the supremum of the ranks of its maximal -ideals. The notion of rank of a maximal -ideal was first introduced and studied in Henriksen et al (1994) and was studied further in Larson (1997). If X is a topological space, and C X denotes the f -ring of all continuous real-valued functions defined on X, then for x ∈ X, we define the rank of x, denoted rank x , to be the rank of the maximal ideal M x = f ∈ C X f x = 0 .…”

“…If X is a topological space, we say X is an SV-space if C X is an SV-ring. SVrings have been studied in Henriksen and Wilson (1992a,b), Henriksen et al (1994), and Larson (To appear). It is shown in Henriksen et al (1994) that all SV spaces have finite rank and it is an open question as to whether the converse holds.…”