# evaluation homomorphism

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 $\psi \mathrm{:}R\mathrm{\to}S$ be a homomorphism^{}. Further, let $s\mathrm{\in}S$. Then there is a unique homomorphism $\varphi \mathrm{:}R\mathit{}\mathrm{[}X\mathrm{]}\mathrm{\to}S$ taking $X$ to $s$ and taking every $r\mathrm{\in}R$ to $\psi \mathit{}\mathrm{(}r\mathrm{)}$.

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 $\varphi (f)$ to be ${\sum}_{i}\psi ({a}_{i}){s}^{i}$. It is clear from the definition of addition^{} and multiplication on polynomials^{} that $\varphi $ is a homomorphism; the definition makes it clear that $\varphi (X)=s$ and $\varphi (r)=\psi (r)$.

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, $\gamma ({a}_{i}{X}^{i})=\psi ({a}_{i}){s}^{i}$ and $\gamma (f)={\sum}_{i}\psi ({a}_{i}){s}^{i}=\varphi (f)$.
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Title | evaluation homomorphism |
---|---|

Canonical name | EvaluationHomomorphism |

Date of creation | 2013-03-22 14:13:51 |

Last modified on | 2013-03-22 14:13:51 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 6 |

Author | mathcam (2727) |

Entry type | Theorem |

Classification | msc 13P05 |

Classification | msc 11C08 |

Classification | msc 12E05 |

Synonym | substitution homomorphism |

Related topic | LectureNotesOnPolynomialInterpolation |

Defines | evaluation homomorphism |