Finite type algebra

Builds on Commutative algebra over a ring Polynomial ring

Definition

Let $A$ be a commutative ring and let $B$ be a commutative $A$ -algebra with structure map $f \colon A \to B$ . The algebra $B$ is of finite type over $A$ , or finitely generated as an $A$ -algebra, if it has a finite generating set. That means there are $b_1, \ldots, b_n \in B$ such that every element of $B$ is $p(b_1, \ldots, b_n)$ for a polynomial $p$ with coefficients in $f(A)$ . Equivalently, there is a surjective $A$ -algebra homomorphism from the polynomial ring $A[x_1, \ldots, x_n]$ onto $B$ for some $n \ge 0$ .

Remarks

Example: $k[x, x^{-1}]$ is of finite type over a field $k$ . It is generated by $x$ and $x^{-1}$ , and it is isomorphic to $k[x, y]/(xy - 1)$ .

Non-example: $\Q$ is not of finite type over $\Z$ . Suppose $\Q = \Z[q_1, \ldots, q_n]$ , and let $d$ be the product of the denominators of the $q_i$ . Then every element of $\Q$ would have a power of $d$ as a denominator, which fails for $1/p$ with $p$ a prime not dividing $d$ .

A finite algebra is of finite type, but $k[x]$ over $k$ shows that the converse fails. A finitely generated algebra over a Noetherian ring is Noetherian, by Hilbert's basis theorem.

Reading path

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Tag	`QM5ZCBJYP1`
URI	`m2://QM5ZCBJYP1`
Kind	definition
Writing	W0 stub
MSC	13E15, 13B02
Defines	Finite type algebra
Uses	Commutative algebra over a ring, Polynomial ring
Changed	2 November 2026

Finite type algebra

Definition

Remarks

Reading path

Further reading

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