IDEALS ASSOCIATED WITH REALCOMPACTNESS IN POINTFREE FUNCTION RINGS

摘要

Let R,L, denote the ring of continuous real-valued functions on a completely regular frame L. The support of an alpha is an element of RL is the closed quotient up arrow(coz alpha)*. We show that if supports are coz-quotients in L, then the set of functions with realcompact support is an ideal. If L satisfies the stronger condition that supports are C-quotients, then this ideal is the intersection of pure parts of the free maximal ideals of R.L. The set of functions whose cozeroes a re realcoinpact is Hi ways ail ideal, which is free if and only if L is locally realcompact if and only if L is (isomorphic an open quotient of vL. Further, this ideal is prime if and only if it is a free real maximal ideal if and only if vL -> L is a one-point extension of L.