---- > [!theorem] Theorem. ([[universal property of polynomial rings]]) > [[polynomial 4|Polynomial rings]] play a role in [[ring]] theory analogous to that of [[free group|free groups]] in [[group]] theory. > > Let $A=\{ a_{1},\dots,a_{n} \}$ be a finite set. Consider the [[category]] $\mathscr{R}^{A}$ whose objects are pairs $(j,R)$ with $R$ a [[commutative ring]] and $j:A \to R$ a set-function and whose morphisms $(j_{1},R_{1}) \to (j_{2}, R_{2})$ > are commutative diagrams of set-functions > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBoBGAXVJADcBDAGwFcYkQBBEAX1PU1z5CKMsWp0mrdgCUA+uR58QGbHgJFypMTQYs2iEHIBMPcTCgBzeEVAAzAE4QAtkjIgcEJJol72AK3lFO0cXRDcPJCMdSX0QAJMaRnoAIxhGAAUBNWEQeywLAAscIJAHZy8aCMQonykDAB16hns0AqxTbiA > \begin{tikzcd} > R_1 \arrow[r, "\varphi"] & R_2 \\ > A \arrow[u, "j_1"] \arrow[ru, "j_2"'] & > \end{tikzcd} > \end{document} > ``` > > in which $\varphi$ is required to be a [[ring homomorphism]]. [^1] Then $(i, \mathbb{Z}[x_{1},\dots, x_{n}])$ is [[terminal object|initial]] in $\mathscr{R}^{A}$, where $i: A \to \mathbb{Z}[x_{1},\dots,x_{n}]$ sends $a_{k}$ to the monomial $x_{k}$. That is, given any [[commutative ring]] $R$ and set-function function $A \xrightarrow{j} R$ there exists a unique [[ring homomorphism]] $\varphi$ making the following diagram commute: > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \usepackage{amssymb} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBoBGAXVJADcBDAGwFcYkQBBEAX1PU1z5CKMsWp0mrdgB1pAW3o4AFgCMVwAFrdkADwD65UgAJZUCDjjH9YSjz4gM2PASKGxNBizaIQAJR7iMFAA5vBEoABmAE4QckhkIDgQSIYSXuxYdpExcYipSUgATB6S3iCyMDpYcBYAhCbSDFFoSpm82bHxNAWIxWlSPgBWIDSM9CowjAAKAs7CIIwwETgB3EA > \begin{tikzcd} > {\mathbb{Z}[x_1, \dots, x_n]} \arrow[r, "\exists! \varphi"] & R \\ > A \arrow[u, "i"] \arrow[ru, "j"'] & > \end{tikzcd} > \end{document} > ``` > ^theorem > [!proof]- Proof. ([[universal property of polynomial rings]]) > **Uniqueness.** To say $\varphi \circ i = j$ is to say $\varphi(x_{k})=j(a_{k})$, so commutativity has enforced $\varphis definition on 'base elements'. Then the fact that it must be a [[ring homomorphism]] enforces its general definition: necessarily, $\begin{align} \varphi\left( \sum_{} m_{i_{1},\dots,i_{n}} x_{1}^{i_{1}} \cdots x_{n}^{i_{n}} \right) =& \sum \varphi(m_{i_{1},\dots,}m_{i_{n}}) \varphi(x_{1})^{i_{1}} \cdots \varphi(x_{n})^{i_{n}} \\ =& \iota(m_{i_{1}}, \dots, m_{i_{n}}) j(a_{1})^{i_{1}} \cdots j(a_{n})^{i_{n}}, \end{align}$ where $\iota:\mathbb{Z} \to R$ is the unique [[ring homomorphism]] from $\mathbb{Z}$ (recall that $\mathbb{Z}$ is [[terminal object|initial]] in $\mathsf{Ring}$). > **Existence.** Easy but omitted, for now. (Though it is where commutativity is useful.) ---- #### [^1]: This should feel like a [[coslice category]]; the difference is that we are mixing objects from $\mathsf{Set}$ and $\mathsf{Ring}$ together, and $\varphi$ is a [[ring homomorphism]] while $j_{1},j_{2}$ are merely set-functions. ----- #### References > [!backlink] > ```dataview TABLE rows.file.link as "Further Reading" FROM [[]] FLATTEN file.tags GROUP BY file.tags as 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 > ```