Category:Definitions/Integral Elements
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.
Pages in category "Definitions/Integral Elements"
The following 5 pages are in this category, out of 5 total.