07 - Universal Property of Quotients

Universal Property of Quotients

Let $\varphi :M\to N$ be a homomorphism of $R$-modules
Let $S$ be a submodule of $M$ with $S\subseteq \ker (\varphi )$

Let $q:M\to M/S$ be quotient homomorphism then

Exists unique homomorphism

$$\overline{\varphi }:M/S\to N$$

such that

$$\varphi =\overline{\varphi }\circ q$$

With $\ker (\overline{\varphi })$ is submodule $\ker (\varphi )/S=\{m+S:m\in \ker (\varphi )\}$

Proof

As $q$ surjective then

$$\overline{\varphi }(q(m))=\varphi (m)$$

uniquely determines values of $\overline{\varphi }$ so $\overline{\varphi }$ is unique if it exists

If $m-m^{\prime }\in S$ then as $S\subseteq \ker (\varphi )$ then

$$0=\varphi (m-m^{\prime })=\varphi (m)-\varphi (m^{\prime })$$

Hence $\varphi$ is constant on $S$-cosets thus induces a map on

$$\overline{\varphi }(m+S)=\varphi (m)$$

As $\overline{\varphi }$ is a homomorphism then
By definition of module structure on quotient $M/S$ then

$$\varphi =\overline{\varphi }\circ q$$

As $\overline{\varphi }(m+S)=\varphi (m)=0$ if and only if $m\in \ker (\varphi )$
Hence

$$m+S\in \ker (\varphi )/S$$