Editing Polynomial ring (section)

=== Categorical characterization ===
{{anchor|free commutative algebra|free commutative ring}}
If {{mvar|K}} is a commutative ring, the polynomial ring {{math|''K''[''X''<sub>1</sub>, …, ''X''<sub>''n''</sub>]}} has the following [[universal property]]: for every [[commutative algebra (structure)|commutative {{mvar|K}}-algebra]] {{mvar|A}}, and every {{mvar|n}}-[[tuple]] {{math|(''x''<sub>1</sub>, …, ''x''<sub>''n''</sub>)}} of elements of {{mvar|A}}, there is a unique [[algebra homomorphism]] from {{math|''K''[''X''<sub>1</sub>, …, ''X''<sub>''n''</sub>]}} to {{mvar|A}} that maps each <math>X_i</math> to the corresponding <math>x_i.</math> This homomorphism is the ''evaluation homomorphism'' that consists in substituting <math>X_i</math> with <math>x_i</math> in every polynomial.

As it is the case for every universal property, this characterizes the pair <math>(K[X_1, \dots, X_n], (X_1, \dots, X_n))</math> up to a unique [[isomorphism]].

This may also be interpreted in terms of [[adjoint functor]]s. More precisely, let {{math|SET}} and {{math|ALG}} be respectively the [[category (mathematics)|categories]] of sets and commutative {{mvar|K}}-algebras (here, and in the following, the morphisms are trivially defined). There is a [[forgetful functor]] <math>\mathrm F: \mathrm{ALG}\to \mathrm{SET}</math> that maps algebras to their underlying sets. On the other hand, the map <math>X\mapsto K[X]</math> defines a functor <math>\mathrm{POL}: \mathrm{SET}\to \mathrm{ALG}</math> in the other direction. (If {{mvar|X}} is infinite, {{math|''K''[''X'']}} is the set of all polynomials in a finite number of elements of {{mvar|X}}.)

The universal property of the polynomial ring means that {{math|F}} and {{math|POL}} are [[adjoint functors]]. That is, there is a bijection
:<math>\operatorname{Hom}_{\mathrm {SET}}(X,\operatorname{F}(A))\cong \operatorname{Hom}_{\mathrm {ALG}}(K[X], A). </math>

This may be expressed also by saying that polynomial rings are '''free commutative algebras''', since they are [[free object]]s 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.