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$M$relativeto$\mathfrak a$andthe$\mathfrak b$-minimum$\mathfrak a$-adjusteddepth$\lambda_{\mathfrak a}^{\mathfrak b}(M)$of$M$, where the underlying module $M$ is relative Cohen-Macaulay w.r.t $\mathfrak{a}$. Some applications of such modules are given.