Module factorisation theorem

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

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

Statement

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

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