Structures determined by prime ideals of rings of functions

Abstract

Introduction.Let 3t and S~z)s respectively denote the category of commutative rings with unity and the category of completely regular Hausdorff spaces; also, denotes the full-subcategory of ¡T^ whose objects are compact, and %>TD the full-subcategory of ^ whose objects are totally-disconnected. The collection of prime ideals of A e ¿% is the underlying set for an object KA e <ßTD and K is contravariantly functorial. If C denotes the contravariant functor which assigns to each Ie5¡¡ the ring C{X) of real-valued continuous functions on X, then the resulting functor Kc is the domain of a natural transformation 8: Kc-> ß, where ß denotes the Stone-Cech reflection of &~3i into &. The prime z-ideals of C{X) also furnish such a space IX, functor £ and natural transformation 8: £ -» ß. In the appropriate category, £ fills in a diagram which exhibits ß as a push-out.Topological properties of KC(X) and l,X are studied and IX is characterized as a certain compactification of XF, the £-topology on X, which helps establish the place of t,X between ßXP and ßX.The above results are applied in an investigation of the continuous and orderpreserved image of £ Y in both of ÇX and ßX arising from/: F-*-X. As one consequence, the prime z-ideal structure and the minimal prime ideal structure associated with X is illuminated by the corresponding structures associated with certain subspaces of X; as another, a convenient simplification and unification is provided for approaching several types of problems found in the literature on

Ideals in the multiplier and corona algebras of a C 0 (X )-algebra

“…The following theorem is due to Montgomery [15]; for a more accessible proof see [22, Theorem 2.1]. The 'patch topology' is defined in [22] but we shall make no use of it.…”