# free commutative algebra

Fix a commutative^{} unital ring $K$ and a set $X$. Then a commutative associative $K$-algebra^{} $F$ is said to be *free on $X$* if there exists an injection $\iota :X\to F$ such that for all functions $f:X\to A$ where $A$ is a commutative $K$-algebra determine a unique algebra homomorphism $\widehat{f}:F\to A$ such that $\iota \widehat{f}=f$. This is an example of a universal mapping property for commutative associative algebras and in categorical settings is often explained with the following commutative diagram^{}:

$$\text{xymatrix}\mathrm{\&}X\text{ar}{[ld]}_{\iota}\text{ar}{[rd]}^{f}\mathrm{\&}F\text{ar}{[rr]}^{\widehat{f}}\mathrm{\&}\mathrm{\&}A.$$ |

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 $X$, and a commutative unital ring $K$, the free commutative associative $K$-algebra on $X$ is the polynomial ring^{} $K\mathit{}\mathrm{[}X\mathrm{]}$.

###### Proof.

Let $A$ be any commutative associative $K$-algebra and $f:X\to A$. Recall $K\u27e8X\u27e9$ is the free associative $K$-algebra on $X$ and so by the universal mapping property of this free object there exists a map $\widehat{f}:K\u27e8X\u27e9\to A$ such that ${\iota}_{K\u27e8X\u27e9}\widehat{f}=f$.

We also have a map $p:K\u27e8X\u27e9\to K[X]$ which effectively
maps words over $X$ to words over $X$. Only in $K[X]$ the
indeterminants commute. Since $A$ is commutative, $\widehat{f}$ factors through
$p$, in the sense that there exists a map $\stackrel{~}{f}:K[X]\to A$ such
that $p\stackrel{~}{f}=\widehat{f}$. Thus $\stackrel{~}{f}$ is the desired map which proves $K[X]$ 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 |