Catenarity in module-finite algebras

The main theorem says that any module-finite (but not necessarily commutative) algebra $\Lambda$ over a commutative Noetherian universally catenary ring $R$ is catenary. Hence the ring $\Lambda$ is catenary if $R$ is Cohen-Macaulay. When $R$ is local and $\Lambda$ is a Cohen-Macaulay $R$-module, we have that $\Lambda$ is a catenary ring, $\dim\Lambda=\dim\Lambda/Q+\mathrm{ht}_\Lambda Q$ for any $Q\in\operatorname{Spec}\Lambda$, and the equality $n=\mathrm{ht}_\Lambda Q- \mathrm{ht}_\Lambda P$ holds true for any pair $P\subseteq Q$ of prime ideals in $\Lambda$ and for any saturated chain $P=P_0\subset P_1\subset \cdots\subset P_n=Q$ of prime ideals between $P$ and $Q$.