Finite morphism of rings

Builds on Ring homomorphism Module over a ring Finitely generated module

Definition

Let $A$ and $B$ be commutative rings and let $f \colon A \to B$ be a ring homomorphism. The ring $B$ is an $A$ -module by $a \cdot b = f(a) b$ . The homomorphism $f$ is finite, and $B$ is a finite $A$ -algebra, if $B$ is a finitely generated $A$ -module. That is, there are $b_1, \ldots, b_n \in B$ such that every $b \in B$ can be written as $b = f(a_1) b_1 + \cdots + f(a_n) b_n$ with $a_i \in A$ .

Remarks

Example: $\Z \to \Z[\sqrt{2}]$ is finite, generated as a module by $1$ and $\sqrt{2}$ . Every surjective homomorphism $A \to A/I$ is finite, generated by $1$ .

Non-example: $\Z \to \Q$ is not finite. If $q_1, \ldots, q_n$ generated $\Q$ as a $\Z$ -module, every rational number would have a denominator dividing the product of their denominators, which fails for $1/p$ with $p$ a large prime.

A homomorphism is finite if and only if it is integral and of finite type. A composition of finite homomorphisms is finite.

Reading path

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Finitely generated module concept
Group concept
Module over a ring concept
Ring definition
Ring homomorphism definition
Finite morphism of rings definition

Tag	`9D7WZEN64H`
URI	`m2://9D7WZEN64H`
Kind	definition
Writing	W0 stub
MSC	13B10, 13B21
Defines	Finite morphism of rings
Uses	Ring homomorphism, Module over a ring, Finitely generated module
Changed	2 November 2026

Finite morphism of rings

Definition

Remarks

Reading path

Further reading

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