Remark

Let M be a submodule of an A-module E and $\phi : E \to E/M$ the canonical homomorphism. For an A-linear mapping $u : E \to F$ to be of the form $v \circ \phi$, where $v$ is a linear mapping of E/M into F, it is necessary and sufficient that $M \subset \mathrm{Ker}(u)$; for, if this condition holds, the relation $x - y \in M$ implies $u(x) = u(y)$, hence $u$ is compatible with this equivalence relation and clearly the mapping $v : E/M \to F$ derived from $u$ by taking quotients is linear.