  The goal of the present paper is the study of some algebraic invariants of
Stanley-Reisner rings of Cohen-Macaulay simplicial complexes of dimension $d -
1$. We prove that the inequality $d \leq \mathrm{reg}(\Delta) \cdot
\mathrm{type}(\Delta)$ holds for any $(d-1)$-dimensional Cohen-Macaulay
simplicial complex $\Delta$ satisfying $\Delta=\mathrm{core}(\Delta)$, where
$\mathrm{reg}(\Delta)$ (resp. $\mathrm{type}(\Delta)$) denotes the
Castelnuovo-Mumford regularity (resp. Cohen-Macaulay type) of the
Stanley-Reisner ring $\Bbbk[\Delta]$. Moreover, for any given integers $d,r,t$
satisfying $r,t \geq 2$ and $r \leq d \leq rt$, we construct a Cohen-Macaulay
simplicial complex $\Delta(G)$ as an independent complex of a graph $G$ such
that $\dim(\Delta(G))=d-1$, $\mathrm{reg}(\Delta(G))=r$ and
$\mathrm{type}(\Delta(G))=t$.
