A book I am reading considers the following setting. Let $A$ be an abelian group and let $Q\subset A$ be a finitely generated submonoid which generates $A$ as a group.
In one argument it is claimed that for a field $k$ the Krull dimension of $k[A]$ equals that of $k[Q]$ with no explanation other than that the embedding $k[Q]\subset k[A]$ is visibly a localization map. What would be a proof of this fact concise enough to be omitted altogether? (I am thus looking for a short proof but, of course, any reasonable proof would be appreciated.)
(I can see that $\dim k[A]\le\dim k[Q]$. I can also see that the Krull dimension of $k[A]$ equals the rank $n$ of the torsion free part of $A$. One could then, probably, show that $k[Q]$ is a module-finite extension of a ring of the form $k[\mathbb Z^l\oplus\mathbb N^{n-l}]$ which would lead to a proof. However, such an approach seems to get messy.)