Polynomials, $\alpha$-ideals, and the principal lattice

Let $R$ be a commutative ring with an identity, $\mathfrak R$ be an almost distributive lattice and $I_\alpha(\mathfrak R)$ be the set of all $\alpha$-ideals of $\mathfrak R$. If $L(R)$ is the principal lattice of $R$, then $R[I_\alpha(\mathfrak R)]$ is Cohen–Macaulay. In particular, $R[I_\alpha(\mathfrak R)][X_1,X_2,\cdots]$ is WB-height-unmixed.