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[parent] evaluation homomorphism (Theorem)

Let $ R$ be a commutative ring and let $ R[X]$ be the ring of polynomials with coefficients in $ R$.

Theorem 1   Let $ S$ be a commutative ring, and let $ \psi\colon R\to S$ be a homomorphism. Further, let $ s\in S$. Then there is a unique homomorphism $ \phi\colon R[X]\to S$ taking $ X$ to $ s$ and taking every $ r\in R$ to $ \psi(r)$.

This amounts to saying that polynomial rings are free objects in the category of $ R$-algebras; the theorem then states that they are projective. This is true in much greater generality; in fact, the property of being projective is intended to extract the essential property of being free.

Proof. We first prove existence. Let $ f\in R[X]$. Then by definition there is some finite list of $ a_i$ such that $ f = \sum_i a_i X^i$. Then define $ \phi(f)$ to be $ \sum_i \psi(a_i) s^i$. It is clear from the definition of addition and multiplication on polynomials that $ \phi$ is a homomorphism; the definition makes it clear that $ \phi(X)=s$ and $ \phi(r)=\psi(r)$.

Now, to show uniqueness, suppose $ \gamma$ is any homomorphism satisfying the conditions of the theorem, and let $ f\in R[X]$. Write $ f = \sum_i a_i X^i$ as before. Then $ \gamma(a_i) = \psi(a_i)$ and $ \gamma(s)$ by assumption. But then since $ \gamma$ is a homomorphism, $ \gamma(a_iX^i) = \psi(a_i)s^i$ and $ \gamma(f) = \sum_i \psi(a_i) s^i = \phi(f)$. $ \qedsymbol$



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See Also: lecture notes on polynomial interpolation

Other names:  substitution homomorphism
Also defines:  evaluation homomorphism

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Attachments:
translation automorphism of a polynomial ring (Example) by archibal
ring adjunction (Definition) by pahio
zero of polynomial (Definition) by pahio
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Cross-references: multiplication, addition, clear, property, category, objects, polynomial rings, coefficients, polynomials, ring, commutative ring
There are 6 references to this entry.

This is version 3 of evaluation homomorphism, born on 2004-03-08, modified 2005-03-18.
Object id is 5671, canonical name is EvaluationHomomorphism.
Accessed 3991 times total.

Classification:
AMS MSC12E05 (Field theory and polynomials :: General field theory :: Polynomials )
 11C08 (Number theory :: Polynomials and matrices :: Polynomials)
 13P05 (Commutative rings and algebras :: Computational aspects of commutative algebra :: Polynomials, factorization)

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