Math 2.0

Commutative algebra Definition

Evaluation homomorphism

Builds on Polynomial evaluationRing homomorphism

Definition

Let RR be a commutative ring and let c∈Rc \in R. The evaluation homomorphism at cc is the map ev⁡c ⁣:R[x]→R,ev⁡c(f)=f(c),\operatorname{ev}_c \colon R[x] \to R, \qquad \operatorname{ev}_c(f) = f(c), where f(c)f(c) is the value of ff at cc. It is a ring homomorphism.

Remarks

The map ev⁡c\operatorname{ev}_c is surjective, because ev⁡c(r)=r\operatorname{ev}_c(r) = r for each constant r∈Rr \in R. Its kernel is the principal ideal (x−c)(x - c). This is a form of the factor theorem. The commutativity of RR is necessary: in general (fg)(c)≠f(c)g(c)(f g)(c) \neq f(c) g(c) when cc does not commute with the coefficients.

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Group, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionPolynomial ring, definition, no evidence levelPolynomial ringdefinitionRing homomorphism, definition, no evidence levelRing homomorphismdefinitionPolynomial evaluation, definition, no evidence levelPolynomial evaluationdefinitionEvaluation homomorphism, definition, no evidence levelEvaluation homomorphi…definition
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Further reading

  1. Web https://en.wikipedia.org/wiki/Polynomial_ring, Polynomial evaluation
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