O.T. O'Meara, A general isomorphism theory for linear groups, 1977.
In case where $D$ is a division ring, both PGL(n, D) and PSL(n, D) remember both $n$ and $D$. Another way of saying the same is that division algebras are uniquely determined by their abstract automorphism groups.
When D is just a subring in division ring, situation becomes a bit trickier.