Suzanne Larson

Lattice-Ordered Algebras That are Subdirect Products of Valuation Domains

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 t-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 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 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.

Pseudoprime L-Ideals in a Class of F-Rings

In a commutative f-ring, an 1-ideal I is called pseudoprime if ab = 0 implies a E I or b E I, and is called square dominated if for every a E I, lal O}, x+ = x V 0, x = (-x) V 0, and [xl = x V (-x). Received by the editors August 21, 1987 and, in revised form, November 20, 1987. This paper was presented to the special session on Ordered Algebraic Structures of the American Mathematical Society on January 8, 1988 in Atlanta, Georgia. 1980 Mathematics Subject Classification (1985 Revision). Primary 06F25. @1988 American Mathematical Society 0002-9939/88

Group Projects and Civic Engagement in a Quantitative Literacy Course

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Rings of Continuous Functions on Spaces of Finite Rank and the SV Property

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Constructing Rings of Continuous Functions in Which There Are Many Maximal Ideals with Nontrivial Rank

Abstract We describe our approach to incorporating a civic engagement component into a quantitative literacy (QL) course and the resulting gains in student learning, confidence, and awareness of local civic issues. We revised the existing QL course by including semester-long group projects involving local community issues that students could investigate using the mathematical topics of the course. Compared to students in the standard QL course, students in the projects-based course performed similarly on all topics on the post-test except one, and significantly better on that one topic. They showed increased awareness of community issues, gained confidence in their ability to respond to mathematical situations using course material, and reported learning non-mathematical skills.

Images and Open Subspaces of SV Spaces

Let X be a compact topological space and let C(X) denote the f-ring of all continuous real-valued functions defined on X. A point x in X is said to have rank n if, in C(X), there are n minimal prime ℓ-ideals contained in the maximal ℓ-ideal M x = {f ∈ C(X):f(x) = 0}. The space X has finite rank if there is an n ∈ N such that every point x ∈ X has rank at most n. We call X an SV space (for survaluation space) if C(X)/P is a valuation domain for each minimal prime ideal P of C(X). Every compact SV space has finite rank. For a bounded continuous function h defined on a cozeroset U of X, we say there is an h-rift at the point z if h cannot be extended continuously to U ∪ {z}. We use sets of points with h-rift to investigate spaces of finite rank and SV spaces. We show that the set of points with h-rift is a subset of the set of points of rank greater than 1 and that whether or not a compact space of finite rank is SV depends on a characteristic of the closure of the set of points with h-rift for each such h. If X has finite rank and the set of points with h-rift is an F-space for each h, then X is an SV space. Moreover, if every x ∈ X has rank at most 2, then X is an SV space if and only if for each h, the set of points with h-rift is an F-space.

The Intermediate Value Theorem for Polynomials over Lattice‐ordered Rings of Functions

Abstract Let X be a topological space, and let C (X) denote the f-ring of all continuous real-valued functions defined on X. For x ∈ X, we define the rank of x to be the number of minimal prime ideals contained in the maximal ideal M x = {f ∈ C (X) : f(x) = 0} if there are finitely many such minimal prime ideals, and the rank of x to be infinite if there are infinitely many minimal prime ideals contained in M x . We call X an SV-space if C (X)/P is a valuation domain for each minimal prime ideal P of C (X). A construction of topological spaces is given showing that a compact SV-space need not be a union of finitely many compact F-spaces and that in a compact space where every point has finite rank, the set of points of rank one need not be open.

Semiprime f -Rings That Are Subdirect Products of Valuation Domains

A topological space is finitely an F-space if its Stone–Čech compactification is a union of finitely many closed F-spaces and a space is SV if C(X) has the property that C(X)/P is a valuation domain for each prime ring ideal P of C(X). This article studies the images under open continuous functions and the open subspaces of spaces that are finitely an F-space or are SV. It is shown that an open continuous image of a compact space that is finitely an F-space is finitely an F-space and an open continuous image of certain SV spaces is SV. Also, it is shown cozerosets, but not necessarily open sets, of SV spaces are SV spaces and a similar situation holds for spaces that are finitely an F-space.

Quasi-Normal f-Rings

The classical intermediate value theorem for polynomials with real coefficients is generalized to the case of polynomials with coefficients in a lattice‐ordered ring that is a subdirect product of totally ordered rings. Several candidates for a generalization are investigated, and particular attention is paid to the case when the lattice‐ordered ring is the algebra C(X) of continuous real‐valued functions on a completely regular topological space X. For all but one of these generalizations, the intermediate value theorem holds only if X is an F‐space in the sense of Gillman and Jerison. Surprisingly, for the most interesting of these generalizations, if X is compact, the intermediate value theorem holds only if X is an F‐space and each component of X is an hereditarily indecomposable continuum. It is not known if there is an infinite compact connected space in which this version of the intermediate value theorem holds.

Sums of Semiprime, Z, and D L-Ideals in a Class of F-Rings

Recall that an f-ring is a lattice-ordered ring in which a Λ b = 0 implies a Λ bc = a Λ cb = 0 whenever c ≥ 0. In [BKW], an f-ring is defined to be a lattice-ordered ring which is a subdirect product of totally ordered rings. These two definitions are equivalent if and only if the prime ideal theorem for Boolean Algebras is assumed; see [FH]. We regard these two definitions as equivalent henceforth. Our main concern is with f-rings that are semiprime; i.e., such that the intersection of the prime ideals is 0. A ring whose only nilpotent element is 0 is said to be reduced. (An f-ring is semiprime if and only if it is reduced; see [BKW, 8.5].) We will, however, maintain more generality when it does not take us too far afield. An l-ideal I of an f-ring A is the kernel of a homomorphism of A into an f-ring. Equivalently, I is a ring ideal of A such that if a ∈ I, b ∈ A, and ∣b∣ < ∣a∣, then b ∈ I. A left ideal with this latter property is called a left l-ideal, and a right l-ideal is defined similarly. We let N(A) denote the set of nilpotent elements of the f-ring A.