02 - Universal Property of Quotients

Universal Property of Quotients

Let $R$ be a ring with ideal $I$ of $R$
Consider Quotient Homomorphism $q:R\to R/I$

If $\varphi :R\to S$ is a ring homomorphism such that

$$I\subseteq \ker (\varphi )$$

then there is unique ring homomorphism

$$\overline{\varphi }:R/I\to S$$

such that

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

and

$$\ker (\overline{\varphi })=\ker (\varphi )/I=\{m+I:m\in \ker (\varphi )\}$$

Proof

Since $q$ is surjective then

$$\overline{\varphi }(q(r))=\varphi (r)\text{ for }r\in R$$

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

If $r-r^{\prime }\in I$ then

$$I\subseteq \ker (\varphi )⟹0=\varphi (r-r^{\prime })=\varphi (r)-\varphi (r^{\prime })$$

Hence $\varphi$ is constant on $I$-cosets

Thus induces map

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

As $\overline{\varphi }$ is a homomorphism following from definition of ring structure on quotient $R/I$

For kernel of $\overline{\varphi }$ then

$$\overline{\varphi }(r+I)=\varphi (r)=0⟺r\in \ker (\varphi )$$

Hence $r+I\in \ker (\varphi )/Iz$