Kernel of a ring homomorphism

Definition

Let $f \colon R \to S$ be a ring homomorphism. The kernel of $f$ is the set $$\ker f = \{a \in R : f(a) = 0\}.$$

Remarks

The kernel of $f$ is a two-sided ideal of $R$. The kernel of the reduction map $\mathbb{Z} \to \mathbb{Z}/n\mathbb{Z}$ is $n\mathbb{Z}$. The map $f$ is injective if and only if $\ker f = \{0\}$, by the kernel criterion.

Explanations

L1 human

The kernel collects all the elements that $f$ sends to $0$. For reduction modulo $5$, the kernel is the set of the multiples of $5$.