First isomorphism theorem for rings

First isomorphism theorem (rings): Let $\varphi:R\to S$ be a ring homomorphism . Then the induced map

is a ring isomorphism , where $\ker(\varphi)$ is the kernel and $\operatorname{im}(\varphi)$ is the image .

This identifies the universal quotient quotient ring determined by $\varphi$ with the concrete subring realized as its image, and is the basic tool behind “modding out by relations” in ring constructions.

Proof sketch: Define $\bar\varphi(r+\ker\varphi)=\varphi(r)$. If $r-r'\in\ker\varphi$, then $\varphi(r)=\varphi(r')$, so the map is well-defined. It is clearly a ring homomorphism, surjective by definition of the image, and injective because $\bar\varphi(r+\ker\varphi)=0$ implies $r\in\ker\varphi$.