---- > [!definition] Definition. ([[free commutative algebra]]) > > > Let $R$ be a [[commutative ring|commutative]] [[ring]] and $A=\{ a_{1},\dots,a_{n} \}$ a finite set. The **free commutative $R$-algebra** on $A$ is the '[[universal property|universal]] [[commutative algebra|commutative]] $R$-[[algebra]] $F(A)$ generated by $A, in a sense captured by the following [[universal property]]: There is a set-function $j:A \to F(A)$ such that any function $f:A \to S$, $S$ a [[commutative algebra|commutative]] $R$-[[algebra]], factors uniquely through $F(A)$ as $f=\varphi \circ j$ via a unique $R$-[[algebra homomorphism]] $\varphi$. This defines $F(A)$ up to [[isomorphism]], if it exists. > > Indeed, it always exists: we have $F(A) \cong R[A]=R[T_{1},\dots T_{n}]$ — formal polynomials in the elements of $A$. > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRACVkBBCkAX1LpMufIRQBGclVqMWbAMr9BIDNjwEiZcdPrNWiEF37SYUAObwioAGYAnCAFskAJmo4ISMjL1sAVkpt7J0QvdyRJbzkDAB1o+ls0AAssAJA7Rxc3D0QI3Si0kGoGOgAjGAYABWF1MRBbLDNEnGM+IA > \begin{tikzcd} > {R[A]} \arrow[r, "\varphi"] & S \\ > A \arrow[u, "j"] \arrow[ru, "f"'] & > \end{tikzcd} > \end{document} > ``` > > > > ^definition (This will feel analogous to parts of [[free module]], [[free abelian group]], [[universal property of polynomial rings]]). Define $j:A \to R[T_{1},\dots,T_{n}]$ as $j(a_{i})=T_{i}$. Consider an arbitrary polynomial $p=\sum_{j}r_{j} T_{1}^{m_{1},j}\cdots T_{n}^{m_{n},j} \in R[T_{1},\dots, T_{n}]$. Because $\varphi$ is an $R$-algebra homomorphism, i.e. a [[ring homomorphism]] + [[linear map]], we have $\begin{align} \varphi(p) &= \sum_{j} r_{j} \big( \varphi(T_{1}^{m_{1}, j} \cdots T_{n}^{m_{n},j}) \big) \tag{$\varphi$ is $R$-linear}\\ &= \sum_{j} r_{j} \big( \varphi(T_{1})^{m_{1}, j} \cdots \varphi(T_{n})^{m_{n}, j} \big) \tag{$\varphi$ is ring homo.} \\ &= \sum_{j} r_{j} f(a_{1}) ^{m_{1}, j} \cdots f(a_{n} ^{m_{n}, j}) \tag{$\varphi(T_{i})= \varphi(j(a_{i}))=f(a_{i})$}. \end{align}$ This determines $\varphi$ uniquely, if it exists. ---- #### - [ ] can also view in terms of [[adjoint functor]], as usual with this flavor of universal property ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```