Title: The finiteness dimension of modules and relative Cohen-Macaulayness,
Abstract: 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.