Let be a commutative ring, and let be a homomorphism. Further, let . Then there is a unique homomorphism taking to and taking every to .
This amounts to saying that polynomial rings are free objects in the category of -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.
We first prove existence. Let . Then by definition there is some finite list of such that . Then define to be . It is clear from the definition of addition and multiplication on polynomials that is a homomorphism; the definition makes it clear that and .
Now, to show uniqueness, suppose is any homomorphism satisfying the conditions of the theorem, and let . Write as before. Then and by assumption. But then since is a homomorphism, and . ∎
|Date of creation||2013-03-22 14:13:51|
|Last modified on||2013-03-22 14:13:51|
|Last modified by||mathcam (2727)|