Kernels are two-sided ideals

Kernels are two-sided ideals: If $\varphi:R\to S$ is a ring homomorphism , then its kernel $\ker(\varphi)$ is a two-sided ideal of $R$.

This lemma is a prerequisite for forming the associated quotient ring $R/\ker(\varphi)$ and is used immediately in the first isomorphism theorem for rings .