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 < x2 for some x E A such that x2 E I. Several characterizations of pseudoprime 1-ideals are given in the class of commutative semiprime frings in which minimal prime 1-ideals are square dominated. It is shown that the hypothesis imposed on the f-rings, that minimal prime 1-ideals are square dominated, cannot be omitted or generalized. Introduction. Let X be a topological space and C(X) be the f-ring of all continuous real-valued functions on X with coordinatewise operations. The following characterizations of pseudoprime i-ideals of C(X) are known. (L. Gillman and C. Kohls [4, 4.1]) For an i-ideal I of C(X), the following are equivalent: (1) I is pseudoprime. (2) The prime ideals containing I form a chain. (3) s/i is prime. In [11], Subramanian asks whether this characterization of pseudoprime i-ideals generalizes to semiprime f-rings. The answer in general is no, as can be seen by Example 2.7. In this work, we investigate pseudoprime i-ideals in the class of commutative semiprime f-rings in which minimal prime i-ideals are square dominated. In this class of f-rings, we give some alternate characterizations of pseudoprime i-ideals, and we show that in normal f-rings conditions (2) and (3) characterize pseudoprime i-ideals. We also show that if all prime i-ideals are square dominated, a generalization of condition (3) characterizes pseudoprime i-ideals in archimedean f-algebras. Finally, we show that the hypothesis imposed on our f-rings, that minimal prime i-ideals be square dominated, cannot be omitted or generalized in any way by showing that if any of the characterizations hold in a semiprime f-ring A, then all minimal prime i-ideals of A are square dominated. We assume throughout that all rings are commutative and semiprime. 1. Preliminaries. An f-ring is a lattice ordered ring which is a subdirect product of totally ordered rings. For general information on f-rings see [2]. Given an f-ring A and x E A, we let A+ = {a E A: a > 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 $1.00 + $.25 per page