On convex ideals

Abstract

In a partially ordered ring, the convex ideals are of special interest because their quotient rings possess a naturally induced order [3, 5.2]. Some partially ordered rings have an abundance of convex ideals; for instance, every prime ideal in the ring of all continuous real-valued functions on a topological space is convex (with respect to the usual order) [3, 5.5]. The results presented here concern those orders on a ring with respect to which every primitive ideal is convex, or every prime ideal is convex. In particular, we give an example of a commutative partially ordered ring with identity in which every maximal ideal is convex but not every prime ideal is convex. A partial order on a ring A can be defined by specifying a set PCA called the positive cone, satisfying the conditions: P+PCP, PPCP, and PO -P = I 0 }. For convenience, we shall identify a partial order on a ring with the positive cone that defines it. An ideal I in A is P-convex if aEP, b-aEP, and b I imply aCI. The ring of all continuous real-valued functions on a topological space X is denoted by C(X). We use the symbol R for the field of real numbers, and Q for the field of rational numbers. The one-point compactification of the countable discrete space N is denoted by N*; the point at infinity is designated by w. The same letter will be used for an ideal I in A and the natural homomorphism of A onto A/I; thus, the image of aeA in A/I is denoted by I(a). Observe that 10 } is a positive cone, and that every ideal in any ring is trivially { O} -convex. It is evident that the union of a chain of positive cones of a ring is again a positive cone, so that Zorn's Lemma implies the existence of a maximal order on any ring. The following statement is also easily verified. For any family 5: of ideals in a ring A, let e be the collection of orders P on A such that every ideal in S; is P-convex. Then e is nonempty, and every order in C is contained in a maximal order in C. In particular, there is a maximal order on A with respect to which every primitive ideal in A is convex, and a maximal order on A with respect to which every prime ideal in A is convex. From now on, we consider mainly commutative semisimple rings. DEFINITION. Let A be a commutative semisimple ring, and let M7 } be the family of primitive (i.e., prime maximal) ideals in A.