First isomorphism theorem for rings

A ring homomorphism induces an isomorphism from the quotient by its kernel onto its image.

First isomorphism theorem for rings

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

\bar\varphi: R/\ker(\varphi)\longrightarrow \operatorname{im}(\varphi),\qquad r+\ker(\varphi)\longmapsto \varphi(r),

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.