(All rings here are always assumed to be unital and associative).

Let $R$ be a ring and $k$ be a commutative subring thereof with the condition that:

if $u$ is a unit in $R$, then $u$ is in $k$.

Can anything be deduced about the global dimension $D(R)$ of $R$, with respect to $D(k)$? I'm strongly inclined to believe, that in such a situation $D(R)\geq D(k)$, for example this hold for $A_n(k)$ and $k[x_1,..,x_n]$.

Moreover, any "counter example" where $S$ is a subring of $R$ and $G(S)\not\leq G(R)$ is generated from an example where $S$ is some unit of $R$, for example any $\mathbb{Z}$-algebra as relating to any $\mathbb{R}$-algebra.