# Equivalence of Definitions of Integral Element of Algebra

## Theorem

The following definitions of the concept of Integral Element of Algebra are equivalent:

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.