Commutative ring/Ring homomorphism/Kernel/Ideal/Fact

Let ${}R$ and ${}S$ be commutative rings, and let

$\varphi \colon R\longrightarrow S$

be a ring homomorphism.

Then the

kernel

${}\operatorname {kern} \varphi ={\left\{f\in R\mid \varphi (f)=0\right\}}\,$

is an ideal

in ${}R$.

Write a proof