Given a ring_homomorphism between rings and , we say the kernel of is the collection of elements of which sends to the zero element of .
Formally, it is where is the zero element of .
Kernels of ring homomorphisms are very important because they are precisely ideals. (Proof.) In a way, "ideal" is to "ring" as "subgroup" is to "group", and certainly subrings are much less interesting than ideals; a lot of ring theory is about the study of ideals.
The kernel of a ring homomorphism always contains , because a ring homomorphism always sends to . This is because it may be viewed as a group homomorphism acting on the underlying additive group of the ring in question, and the image of the identity is the identity in a group.
If the kernel of a ring homomorphism contains , then the ring homomorphism sends everything to . Indeed, if , then .