On a class of subalgebras of $C(X)$ and the intersection of their free maximal ideals

Abstract

Abstract. Let X be a Tychonoff space and A a subalgebra of C(X) containing C * (X). Suppose that C K (X) is the set of all functions in C(X) with compact support. Kohls has shown that C K (X) is precisely the intersection of all the free ideals in C(X) or in C * (X). In this paper we have proved the validity of this result for the algebra A. Gillman and Jerison have proved that for a realcompact space X, C K (X) is the intersection of all the free maximal ideals in C(X). In this paper we have proved that this result does not hold for the algebra A, in general. However we have furnished a characterisation of the elements that belong to all the free maximal ideals in A. The paper terminates by showing that for any realcompact space X, there exists in some sense a minimal algebra Am for which X becomes Am-compact. This answers a question raised by Redlin and Watson in 1987. But it is still unsettled whether such a minimal algebra exists with respect to set inclusion.

Abundance of Isomorphic and non isomorphic intermediate rings

“…The structure space of a commutative ring R is the set of all maximal ideals of R with hull kernel topology. Intermediate rings have been studied by several authors, viz., [3], [2], [4], [8], [9], [10]. It is perhaps well known, though we shall give a short proof of it of the fact that if X is not pseudocompact meaning that C * (X) C(X), then (X) contains at least 2 c many distinct rings.…”