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.
Theorem 1.
Given a set , and a commutative unital ring , the free commutative associative -algebra on is the polynomial ring .
Proof.
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.
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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 |