Definition:Integral Element of Algebra/Definition 4
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Definition
Let $A$ be a commutative ring with unity.
Let $f : A \to B$ be a commutative $A$-algebra.
Let $b\in B$.
The element $b$ is integral over $A$ if and only if there exists a faithful $A \sqbrk b$-module whose restriction of scalars to $A$ is finitely generated.