Math 2.0

Associative rings and algebras Definition

Kernel of a ring homomorphism

Builds on Ring homomorphism

Definition

Let f ⁣:R→Sf \colon R \to S be a ring homomorphism. The kernel of ff is the set ker⁡f={a∈R:f(a)=0}.\ker f = \{a \in R : f(a) = 0\}.

Remarks

The kernel of ff is a two-sided ideal of RR. The kernel of the reduction map ℤ→ℤ/nℤ\mathbb{Z} \to \mathbb{Z}/n\mathbb{Z} is nℤn\mathbb{Z}. The map ff is injective if and only if ker⁡f={0}\ker f = \{0\}, by the kernel criterion.

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Group, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionRing homomorphism, definition, no evidence levelRing homomorphismdefinitionKernel of a ring homomorphism, definition, no evidence levelKernel of a ring homo…definition
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Further reading

  1. Web https://en.wikipedia.org/wiki/Kernel_(algebra), Ring homomorphisms
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