Let $A$ be a commutative ring, $B$ be a non-commutative $A$-algebra, $P$ be a finitely generated projective left $B$-module.
Let $E=\text{End}_B(P)$ as a left $A$-module (by restricting from $B$). If I were to know that $E$ is in fact a free $A$-module (since $A$, and the map $f:A\to B$ defining the $A$-algebra structure are both particularly nice, say), how would I go about determining the rank of $E$? Say that I know $\text{Spec}(A)$ is connected, so that $P$ (after restriction to being an $A$-module) has well defined rank $r$ (where $r$ is known), and I know all of its generators of $P$ as a $B$-module.
I asked a tangentially related question here. I've also tried to approach this a few ways. In particular I was originally hoping I could use that $P\oplus Q\cong B^n$, for some $n$ and some other $B$-module $Q$, and $\text{End}_B(P\oplus Q)\cong M_n(B^{\text{op}})$ but couldn't work out how to make this useful, since I need to restrict to an $A$-module.