Category:Definitions/Integral Elements

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This category contains definitions related to Integral Elements.
Related results can be found in Category:Integral Elements.


Let $A$ be a commutative ring with unity.

Let $f : A \to B$ be a commutative $A$-algebra.

Let $b \in B$.


Definition 1

The element $b$ is integral over $A$ if and only if it is a root of a monic polynomial in $A \sqbrk x$.


Definition 2

The element $b$ is integral over $A$ if and only if the generated subalgebra $A \sqbrk b$ is a finitely generated module over $A$.


Definition 3

The element $b$ is integral over $A$ if and only if the generated subalgebra $A \sqbrk b$ is contained in a subalgebra $C \le B$ which is a finitely generated module over $A$.


Definition 4

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.