---- (Everything here is [[commutative ring|commutative]], and we focus on finite generating sets.) > [!definition] Definition. ([[subalgebra generated by a subset]]) > Let $R$ be a [[ring]] and $A$ an $R$-[[algebra]]. Let $\{ a_{1},\dots,a_{n} \}$ be a finite subset of $A$. The [[free commutative algebra|universal property of free algebras]] produces a unique [[algebra homomorphism|homomorphism of]] $R$-[[algebra|algebras]] $\varphi:R[T_{1},\dots,T_{n}] \to A$ > from the [[polynomial 4|polynomial algebra]] $R[T_{1},\dots,T_{n}]$ to $A$. > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRACVkAVAfQEZSAHUFQIOOKQAEvMBRABfUuky58hFAMrV6zVohABBBUpAZseAkQF8qtRizbCAxkzTHl5tVdIAmWzod9YWA6filhUXEpULBheQVbGCgAc3giUAAzACcIAFskAGZqHAgkARAAIxgwKCQAWgLiRUyc-MQikBKkJpNsvO7i0sRygL0QYRgADyw4cUkAQklheiy0AAssBPkgA > \begin{tikzcd} > {R[T_1,\dots, T_n]} \arrow[r, "\exists ! \varphi"] & A \\ > & \cup \\ > & {\{a_1, \dots, a_n\}} \arrow[uu, bend right] \arrow[luu] > \end{tikzcd} > \end{document} > ``` > $\im \varphi$ is an $R$-[[algebra|subalgebra]] of $A$, called the **subalgebra generated by $\{ a_{1},\dots,a_{n} \}$** and denoted (perhaps confusingly) $R[a_{1},\dots,a_{n}]$. > > Its elements look like $R$-combinations of products of powers of $a_{1},\dots,a_{n}$: $\im \varphi=\left\{ \sum_{j} r_{j} a_{1}^{\alpha_{1},j} \cdots a_{n}^{\alpha_{n},j} \ : \ r_{j} \in R, \alpha_{k,j} \geq 0 \right\} \ (*)$ > If $\varphi$ is a [[surjection]] — meaning that *any* $a \in A$ can be written in such a way — then we say $A$ is **finitely generated as an algebra over $R$**, or that $A$ is **of finite type as an $R$-algebra**. In other words, $A$ is an $R$-algebra finite-type if and only if it arises as a quotient of $R[T_{1},\dots,T_{n}]$ for some $n$. > [!basicexample] Example. > Any [[finite algebra|finite]] $R$-[[algebra]] is of finite type: its elements are all (finite) linear combinations $\sum_{j=1}^{n}r_{j}a_{j}$ and hence of the form in $(*)$. > > > [!basicnonexample] The converse is clearly false! > > The [[polynomial 4|polynomial ring]] $R[x]$ is a finite-type $R$-algebra, but it is not [[finite algebra|finite]] (not [[submodule generated by a subset|finitely generated]] as an $R$-[[module]]). > If we assume that our $R$-[[algebra|algebras]] are [[integral algebra|integral]], then the notions agree: see [[integral algebra]]. ^nonexample ---- #### [^1]: Beware a notation clash with $A$ in the note [[free commutative algebra]]. ---- #### 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 > ```