Definition:Finite Ring Homomorphism

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

Then is a $\phi$ finite (ring) homomorphism if there exists a finite number of $b_1, \ldots, b_n$ such that every $b \in B$ can be written as:


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

where $a_i \in A$.

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