\(f\)-rings in which every maximal ideal contains finitely many minimal prime ideals. (English) Zbl 0952.06026

Larson, Suzanne

Commun. Algebra 25, No. 12, 3859-3888 (1997).

From the introduction: In this paper, all \(f\)-rings are assumed to be commutative, semiprime, and to have an identity element. We will study \(f\)-rings of finite rank. In the class of commutative semiprime \(f\)-rings with identity element, we give several characterizations of \(f\)-rings of finite rank. We also show that it is possible for commutative semiprime \(f\)-rings with identity element to have the property that every maximal \(\ell\)-ideal has finite rank while the \(f\)-ring does not have finite rank. Rings of continuous functions of finite rank are studied. Some sufficient conditions are given under which a topological space \(X\) such that every fixed maximal \(\ell\)-ideal of \(C(X)\) has finite rank is an SV-space.
A commutative \(f\)-ring with identity element is said to be quasi-normal if the sum of any two different minimal prime \(\ell\)-ideals is either a maximal \(\ell\)-ideal or the entire \(f\)-ring. In our final section, we give characterizations of normal spaces \(X\) for which the corresponding ring of continuous functions is both quasi-normal and SV.

MSC:

06F25	Ordered rings, algebras, modules
54C40	Algebraic properties of function spaces in general topology
54D15	Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)

Keywords:

SV-ring; \(f\)-rings of finite rank; semiprime \(f\)-rings; \(\ell\)-ideal; SV-space; commutative \(f\)-ring; quasi-normal; normal spaces; ring of continuous functions

References:

[1]	Bigard A., Groupes et Anneaux Reticules (1977) · Zbl 0384.06022
[2]	Cherlin G., Fundamenta Math 126 pp 147– (1986)
[3]	Darnel M. R., Theory of Lattice-Ordered Groups (1994) · Zbl 0810.06016
[4]	Gillman L., Rings of Continuous Functions (1960) · Zbl 0093.30001
[5]	Henriksen M., Proceedings of the Conference on Ordered Algebraic Structures - Gainesville pp 159– (1993) · doi:10.1007/978-94-011-1723-4_10
[6]	Henriksen M., Trans. Amer. Math. Soc 345 pp 195– (1994) · Zbl 0817.06014 · doi:10.2307/2154601
[7]	Henriksen M., Topol. and Appl 44 pp 175– (1992) · Zbl 0801.54014 · doi:10.1016/0166-8641(92)90091-D
[8]	Henriksen M., Topol. and Appl 46 pp 89– (1992) · Zbl 0791.54049 · doi:10.1016/0166-8641(92)90123-H
[9]	Huckaba J., Commutative Rings with Zero Divisors (1988) · Zbl 0637.13001
[10]	Jacobson N., Basic Algebra II (1980) · Zbl 0441.16001
[11]	Larson S., Canad. J. Math 38 pp 48– (1986) · Zbl 0588.06011 · doi:10.4153/CJM-1986-003-6
[12]	Larson S., Comm. in Algebra 23 pp 5461– (1995) · Zbl 0847.06007 · doi:10.1080/00927879508825545
[13]	Larson S., Ordered Algebraic Structures, Kluwer Academic Publishers 23 pp 261– (1997)
[14]	Martinez J., in Algebra 20 pp 2975– (1992) · Zbl 0766.06018 · doi:10.1080/00927879208824500
[15]	Subramanian H., Bull. soc. Math. Prance 95 pp 193– (1967)