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: $\tilde{B}\subseteq \widehat{B}$ for every affine semigroup $\widehat{B}$ with $B\subseteq \widehat{B}\subseteq C(B)$ such that $K[\hat B]$ is Cohen-Macaulay.