Ker­nel of ring homomorphism

WikiLast edit: 4 Aug 2016 19:38 UTC by Patrick Stevens

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 .

Examples

Properties

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 .

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