free commutative algebra
Fix a commutative unital ring and a set . Then a commutative associative -algebra is said to be free on if there exists an injection such that for all functions where is a commutative -algebra determine a unique algebra homomorphism such that . This is an example of a universal mapping property for commutative associative algebras and in categorical settings is often explained with the following commutative diagram:
To construct a free commutative associative algebra we observe that commutative associative algebras are a subcategory of associative algebras and thus we can make use of free associative algebras in the construction and proof.
Given a set , and a commutative unital ring , the free commutative associative -algebra on is the polynomial ring .
Let be any commutative associative -algebra and . Recall is the free associative -algebra on and so by the universal mapping property of this free object there exists a map such that .
We also have a map which effectively maps words over to words over . Only in the indeterminants commute. Since is commutative, factors through , in the sense that there exists a map such that . Thus is the desired map which proves is free in the category of commutative associative algebras. ∎
|Title||free commutative algebra|
|Date of creation||2013-03-22 16:51:22|
|Last modified on||2013-03-22 16:51:22|
|Last modified by||Algeboy (12884)|
|Defines||free commutative algebra|