[summary: In ring theory, the notion of "[ideal_ring_theory ideal]" corresponds precisely with the notion of "kernel of [-ring_homomorphism]".]

In ring theory, the notion of "[ideal_ring_theory 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 %%note:That is, after removing the multiplicative structure from the ring.%% 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 $f(kr) = f(k)f(r) = 0 \times r = 0$, so $kr$ is in $K$ if $k$ is.

Ideals are kernels

[todo: refer to the quotient group, and therefore introduce the quotient ring]