Definition:Finite Ring Homomorphism

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

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.