The finiteness dimension of modules and relative Cohen-Macaulayness

Let \(R\) be a commutative Noetherian ring, \(\mathfrak a\) and \(\mathfrak b\) ideals of \(R\). In this paper, we study the finiteness dimension \(f_{\mathfrak a}(M)\) of \(M\) relative to \(\mathfrak a\) and the \(\mathfrak b\)-minimum \(\mathfrak a\)-adjusted depth \(\lambda_{\mathfrak a}^{\mathfrak b}(M)\) of \(M\), where the underlying module \(M\) is relative Cohen-Macaulay w.r.t \(\mathfrak a\). Some applications of such modules are given.