We just introduced the notion of the Krull dimension of a ring in class and I was thinking about the following:
Let A be a commutative, noetherian ring with unit and let $n=\text{dim}(A)$ be the Krull dimension. Furthermore, let $M$ be a free, commutative monoid. Are there any circumstances under which the following holds? \begin{align*} \mathrm{dim}(A[M])=\text{dim}(A) + \text{rk}(M) \end{align*} Maybe if we assume $M$ to be finitely generated and torsion-free? I have tried looking for something similar but I have not found anything yet. If this is never true I would be glad if anyone could provide some counter-examples.