Definition:Finite morphism of rings

Definition

Let $\varphi : A \to B$ be a morphism of rings.

We call $\varphi$ finite if there exists a finite number of $b_1, \ldots, b_n$ such that every $b \in B$ can be written as:

$\displaystyle b = \sum_{i=1}^n \varphi \left({a_i}\right) b_i$

where $a_i \in A$.

Alternatively, $B $ is an $A$-module through $\varphi$, and $\varphi$ is called finite if $B$ is a finitely generated $A$-module.