In ring theory, the notion of "ideal" corresponds precisely with the notion of "kernel of ring_homomorphism".

This result is analogous to the fact from group theory that normal subgroups are the same thing as kernels of group homomorphisms (proof).

Proof

Kernels are ideals

Let $f:R\to S$ be a ring homomorphism between rings $R$ and $S$. We claim that the kernel $K$ of $f$ is an ideal.

Indeed, it is clearly a subgroup of the ring $R$ when viewed as just an additive group ^[1] because $f$ is a group homomorphism between the underlying additive groups, and kernels of group homomorphisms are subgroups (indeed, normal subgroups). (Proof.)

We just need to show, then, that $K$ is closed under multiplication by elements of the ring $R$. But this is easy: if $k\in K$ and $r\in R$, then $f\left(kr\right)=f\left(k\right)f\left(r\right)=0\times r=0$, so $kr$ is in $K$ if $k$ is.

Ideals are kernels

1. ^{^︎}

   That is, after removing the multiplicative structure from the ring.