[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).
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]