Universal property of quotient modules

Universal property of quotient modules: Let $M$ be an $R$-module, let $N\le M$ be a submodule, and let $\pi:M\to M/N$ be the quotient map. For any $R$-module $Q$ and any homomorphism $f:M\to Q$ such that $N\subseteq \ker(f)$, there exists a unique homomorphism $\bar f:M/N\to Q$ with $f=\bar f\circ \pi$.

This is the defining mapping property of the quotient module , and it expresses quotients as the universal way to force a submodule to lie in the kernel of a module homomorphism .