Category:Integral Elements

This category contains results about Integral Elements.
Definitions specific to this category can be found in Definitions/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.