Definition:Finite Ring Homomorphism

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

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

Definition 1
$\phi$ is finite $B$ is finite as an algebra over $A$ via $\phi$.

Definition 2
$\phi$ is finite 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 $a_i \in A$.

Also see

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