Computation of depth of factor rings of C(X)

Abstract

It is known that the depth of every factor ring of [Formula: see text] module an ideal is at most 1. In this paper, we examine conditions under which the depth of factor rings of [Formula: see text] module closed ideals are either 0 or 1. Particularly, we show that the depth of factor ring [Formula: see text], [Formula: see text], is 0 (or equivalently this ring is classical i.e. its every element is unit or zerodivisor) if and only if [Formula: see text] is an almost [Formula: see text]-space completely separated from every zero-set disjoint from it. Using this, it has been confirmed that [Formula: see text] modulo the smallest [Formula: see text]-ideal containing [Formula: see text] is classical if and only if [Formula: see text] is an almost [Formula: see text]-space completely separated from every zero-set disjoint from it. Also, it has been verified that [Formula: see text] is a [Formula: see text]-space if and only if for every ideal [Formula: see text], the factor ring [Formula: see text] has depth zero. Finally, we present a counterexample to a conjecture about the depth of subrings of [Formula: see text] in the literature.