free commutative algebra

Fix a commutativePlanetmathPlanetmathPlanetmath unital ring K and a set X. Then a commutative associative K-algebraMathworldPlanetmathPlanetmathPlanetmath F is said to be free on X if there exists an injection ι:XF such that for all functions f:XA where A is a commutative K-algebra determine a unique algebra homomorphism f^:FA such that ιf^=f. This is an example of a universal mapping property for commutative associative algebras and in categorical settings is often explained with the following commutative diagramMathworldPlanetmath:


To construct a free commutative associative algebra we observe that commutative associative algebras are a subcategoryMathworldPlanetmath of associative algebras and thus we can make use of free associative algebras in the construction and proof.

Theorem 1.

Given a set X, and a commutative unital ring K, the free commutative associative K-algebra on X is the polynomial ringMathworldPlanetmath K[X].


Let A be any commutative associative K-algebra and f:XA. Recall KX is the free associative K-algebra on X and so by the universal mapping property of this free object there exists a map f^:KXA such that ιKXf^=f.

We also have a map p:KXK[X] which effectively maps words over X to words over X. Only in K[X] the indeterminants commute. Since A is commutative, f^ factors through p, in the sense that there exists a map f~:K[X]A such that pf~=f^. Thus f~ is the desired map which proves K[X] is free in the categoryMathworldPlanetmath of commutative associative algebras. ∎

Title free commutative algebra
Canonical name FreeCommutativeAlgebra
Date of creation 2013-03-22 16:51:22
Last modified on 2013-03-22 16:51:22
Owner Algeboy (12884)
Last modified by Algeboy (12884)
Numerical id 5
Author Algeboy (12884)
Entry type Theorem
Classification msc 08B20
Related topic PolynomialRing
Defines free commutative algebra