Factorisation of a map of modules

Question

Suppose you have a ring $A$, $B$ with and an $A$-module $M$. Suppose the projection $q:A\rightarrow B$ is surjective, and that there's a map $f:A\rightarrow M$. Under what assumptions and why does $f$ factor through a map $B\rightarrow M$? $$ \newcommand{\ra}[1]{\!\!\!\!\!\!\!\!\!\!\!\!\xrightarrow{\quad#1\quad}\!\!\!\!\!\!\!\!} \newcommand{\da}[1]{\left\downarrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.} % \begin{array}{lllllll} A & \ra{f} & M \\ \da{q} & & \\ B & \\ \end{array} $$ Sorry for the shitty diagram, but basically the question is when is there a unique arrow from $B$ to $M$ that makes the diagram commute.