Natural Cohen-Macaulayfication of simplicial affine semigroup rings

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.