Kernel is an ideal

Kernel is an ideal: Let $\varphi:R\to S$ be a ring homomorphism. Then

is a two-sided ideal of $R$.

For a ring homomorphism $\varphi$, the kernel is therefore an ideal (indeed a two-sided ideal in general), so one can form the quotient ring $R/\ker\varphi$. This is the key input for the First Isomorphism Theorem for rings .