Intermediate rings of complex-valued continuous functions

<p>For a completely regular Hausdorff topological space X, let C(X, C) be the ring of complex-valued continuous functions on C ∗ (X, its subring bounded functions, and Σ(X, denote collection all rings that lie between C). We show there is natural correlation absolutely convex ideals/ prime ideals/maximal ideals/z-ideals/z ◦ -ideals in P(X, their real-valued counterparts ∩ C(X). These correlations culminate to fact structure any such βX. For ideal I C), we observe C)+I member which further isomorphic type C(Y, Incidentally these are only C-type intermediate if X pseudocompact. for maximal M C)/M an algebraically closed field, furthermore algebraic closure C(X)/M ∩C(X). give necessary sufficient condition CP consists those whose support P sets ideal, examine few special cases thereafter. At end article, find estimates standard parameters concerning zero-divisor graphs C).</p>