Polynomial ring
Categorical characterization

If K is a commutative ring, the polynomial ring K has the following universal property: for every commutative K -algebra A, and every n - tuple (x, …, x) of elements of A, there is a unique algebra homomorphism from K to A that maps each X i {\displaystyle X_{i}} to the corresponding x i. {\displaystyle x_{i}.} This homomorphism is the evaluation homomorphism that consists in substituting X i {\displaystyle X_{i}} with x i {\displaystyle x_{i}} in every polynomial.

As it is the case for every universal property, this characterizes the pair (K, (X 1, …, X n)) {\displaystyle (K,(X_{1},\dots,X_{n}))} up to a unique isomorphism.

This may also be interpreted in terms of adjoint functors. More precisely, let SET and ALG be respectively the categories of sets and commutative K -algebras (here, and in the following, the morphisms are trivially defined). There is a forgetful functor F: A L G → S E T {\displaystyle \mathrm {F} :\mathrm {ALG} \to \mathrm {SET} } that maps algebras to their underlying sets. On the other hand, the map X ↦ K {\displaystyle X\mapsto K} defines a functor P O L: S E T → A L G {\displaystyle \mathrm {POL} :\mathrm {SET} \to \mathrm {ALG} } in the other direction. (If X is infinite, K is the set of all polynomials in a finite number of elements of X.)

The universal property of the polynomial ring means that F and POL are adjoint functors. That is, there is a bijection

This may be expressed also by saying that polynomial rings are free commutative algebras, since they are free objects in the category of commutative algebras. Similarly, a polynomial ring with integer coefficients is the free commutative ring over its set of variables, since commutative rings and commutative algebras over the integers are the same thing.
change_case:english_lowercase