  Let $K$ be a field, $B$ a simplicial affine semigroup, and $C(B)$ the
corresponding cone. We will present a decomposition of $K[B]$ into a direct sum
of certain monomial ideals, which generalizes a construction by Hoa and
St\"uckrad. We will use this decomposition to construct a semigroup $\tilde B$
with $B \subseteq \tilde B \subseteq C(B)$ such that $K[\tilde B]$ is
Cohen-Macaulay with the property: $\tilde B \subseteq \hat B$ for every affine
semigroup $\hat B$ with $B \subseteq \hat B \subseteq C(B)$ such that $K[\hat
B]$ is Cohen-Macaulay.
