Variations on a Theorem on Rings of Continuous Functions

Abstract

We find variations of the classical theorem that any nonzero real ring homomorphism, $u$, of $C(X)$, for a compact Hausdorff space, $X$, is fixed. We let $X$ be a locally compact Hausdorff space and we let $u$ be defined on certain subrings of $C(X)>$ We also vary the hypothesis on $u$ in other ways.