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

Let $R$ be a ring, and $A$ and $B$ be $R$-algebras, with $A$ a commutative subalgebra of $B$ statisfying:

*if $u$ is a unit in $B$, then $u$ is in $A$.* 

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

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