Constructing rings of continuous functions in which there are many maximal ideals with nontrivial rank. (English) Zbl 1024.54015

Larson, Suzanne

Commun. Algebra 31, No. 5, 2183-2206 (2003).

If \(M\) is a maximal \(\ell\)-ideal of the \(f\)-ring \(A\), then the rank of \(M\) is the number of minimal prime ideals contained in \(M\). The rank of an \(f\)-ring is the supremum of the ranks of its maximal \(\ell\)-ideals. A commutative ring is a valuation domain if it has no proper divisors of zero and given any pair of elements, one divides the other. An SV-ring is a commutative \(f\)-ring \(A\) such that \(A/P\) is a valuation domain for each prime ring ideal \(P\) of \(A\). Say that a (Tychonoff) topological space \(X\) is an SV-space if the ring \(C(X)\) of continuous real valued functions on \(X\) is an \(SV\)-ring and the rank of \(X\) is the rank of \(C(X)\). Finally define the rank of \(x\in X\) to be the rank of the maximal ideal \(M_x=\{f\in C(X):f(x)=0\}\). All previously known examples of compact \(SV\)-spaces and compact spaces of finite rank are finite unions of compact \(F\)-spaces (that is to say, spaces in which every finitely generated ring ideal of \(C(X)\) is principal) and have a dense open subspace of points of rank 1. The main purpose of this paper is to construct a compact \(SV\)-space which is not the union of a finite number of compact \(F\)-spaces and in which the set of points of rank 1 is not open. The construction is ingenious but somewhat complicated.

Reviewer: Richard G.Wilson (México)

MSC:

54C40	Algebraic properties of function spaces in general topology
06F25	Ordered rings, algebras, modules
54C35	Function spaces in general topology
46E25	Rings and algebras of continuous, differentiable or analytic functions

Keywords:

\(SV\)-space; \(F\)-space; \(f\)-ring; rank of a maximal ideal; rank of a point

References:

[1]	Atiyah M. F., Introduction to Commutative Algebra (1969) · Zbl 0175.03601
[2]	Bigard A., Lecture Notes in Math. 608, in: Groupes et Anneaux Réticulés (1977) · Zbl 0384.06022
[3]	Dugundji J., Topology (1966)
[4]	Fine N. J., Rings of Quotients of Rings of Functions (1965)
[5]	Gillman L., Rings of Continuous Functions (1960) · Zbl 0093.30001
[6]	DOI: 10.1016/0166-8641(92)90091-D · Zbl 0801.54014 · doi:10.1016/0166-8641(92)90091-D
[7]	DOI: 10.1016/0166-8641(92)90123-H · Zbl 0791.54049 · doi:10.1016/0166-8641(92)90123-H
[8]	DOI: 10.2307/2154601 · Zbl 0817.06014 · doi:10.2307/2154601
[9]	DOI: 10.4153/CJM-1986-003-6 · Zbl 0588.06011 · doi:10.4153/CJM-1986-003-6
[10]	DOI: 10.1080/00927879708826092 · Zbl 0952.06026 · doi:10.1080/00927879708826092
[11]	DOI: 10.1080/00927879208824500 · Zbl 0766.06018 · doi:10.1080/00927879208824500