Definition:Finite Ring Homomorphism

This page has been identified as a candidate for refactoring of basic complexity. In particular: multiple definitions Until this has been finished, please leave {{Refactor}} in the code. New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only. Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Refactor}} from the code.

There are no source works cited for this page. Source citations are highly desirable, and mandatory for all definition pages. Definition pages whose content is wholly or partly unsourced are in danger of having such content deleted. To discuss this page in more detail, feel free to use the talk page.

Definition

Let $A$ and $B$ be commutative rings with unity.

Let $\phi: A \to B$ be a ring homomorphism.

Definition 1

$\phi$ is finite if and only if $B$ is finite as an algebra over $A$ via $\phi$.

Definition 2

$\phi$ is finite if and only if there exists a finite number of $b_1, \ldots, b_n$ such that every $b \in B$ can be written as:

$\ds b = \sum_{i \mathop = 1}^n \map \phi {a_i} b_i$

where $a_i \in A$.

Also see

• Equivalence of Definitions of Finite Ring Homomorphism
• Definition:Finite Morphism of Schemes