LAUSR

Abstract We show that every regular sequence in C ( X ) has length ≤1. This shows that depth ( C ( X ) ) ≤ 1 . We also show that the depth of each maximal ideal of C ( X ) is either zero or one. In fact we observe that X is an almost P-space if and only if the depth of each maximal ideal of C ( X ) is zero and X contains at least one non-almost P-point if and only if depth ( M ) = 1 for each maximal ideal M of C ( X ) . Using this it turns out that for a given topological space X, there are no maximal ideals in C ( X ) with different depths. Regular sequences are also investigated in the factor rings of C ( X ) and we observe that such sequences in the factor rings of C ( X ) modulo principal z-ideals have also length ≤1. Finally, we obtain some topological conditions for which the depth of factor rings of C ( X ) modulo some closed ideals are zero.