All rings are assumed to be associative and unital.
If $B$ is a commutative sub-ring of $A$ (which itself needs not be commutative) then what properties of $B$ are both necessary and sufficient for $D(B)\leq D(A)$ (where $D(-)$ denotes the global dimension)?
For example: the property: $B$ is in the centre of $A$ and $B$ is a free as an $A$ module is sufficient for $D(A)\leq D(B)$ , but is there a tighter result?