Definition:Finite Ring Homomorphism

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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 $a_i \in A$.

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