Module factorisation theorem

{{Stub page|grade=A*|msg=Todo:

1. Quotient module
2. Tidy up
3. Add proof

Statement

Let $(R,+,*,0)$ be a ring (with or without unity) and let $M$ be a (left) [[R-module|$R$-module}}. Let $A$ be a submodule of $A$. Then:

• for every homomorphism $\varphi:M\rightarrow B$ for some $R$-module $B$ whose kernel contains $A$
  • $\varphi$ factors uniquely though the canonical projection $\pi:M\rightarrow \frac{M}{A}$
  • That is to say there is a unique $\psi:\frac{M}{A}\rightarrow B$ such that $\varphi=\psi\circ\pi$