Ideals are the same thing as kernels of ring homomorphisms

https://arbital.com/p/ideal_equals_kernel_of_ring_homomorphism

by Patrick Stevens Aug 3 2016


[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 be a ring homomorphism between rings and . We claim that the kernel of is an ideal.

Indeed, it is clearly a Subgroup of the ring when viewed as just an additive group %%note:That is, after removing the multiplicative structure from the ring.%% because 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 is closed under multiplication by elements of the ring . But this is easy: if and , then , so is in if is.

Ideals are kernels

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