Computation of depth of factor rings of \(C(X)\) (Q6608208)

If \(M\neq 0\), the supremum of integers \(n\) (if it exists) such that there is an \(R\)-sequence of length \(n\) on \(M\) is called the depth of \(M\), denoted by \(\text{depth}(M)\).\N\NIn this paper, the authors compute the depth of the factor rings of \(C(X)\), i.e., the ring of all real-valued continuous functions on a completely regular Hausdorff (Tychonoff) space \(X\), modulo (some) closed ideals. For example, it is shown that for a non-empty subspace \(A\) of \(X\), the depth of the factor ring \(\frac{C(X)}{M_A}\) is zero if and only if \(A\) is an almost P-space (i.e., every non-empty zero-set in \(A\) has non-empty interior) which is completely separated from every disjoint zero-set. Furthermore, the authors prove that \(X\) is a \(P\)-space (i.e., every zero-set of \(X\) is open) if and only if for each semiprime ideal \(I\) of \(C(X)\), the depth of the factor ring \(\frac{C(X)}{I}\) equals zero.