Equivalence of Definitions of Associate in Integral Domain/Definition 1 Equivalent to Definition 2

Theorem

The following definitions of the concept of Associate in the context of Integral Domain are equivalent:

Let $\struct {D, +, \circ}$ be an integral domain.

Let $x, y \in D$.

Definition 1

$x$ is an associate of $y$ (in $D$) if and only if they are both divisors of each other.

That is, $x$ and $y$ are associates (in $D$) if and only if $x \divides y$ and $y \divides x$.

Definition 2

$x$ and $y$ are associates (in $D$) if and only if:

$\ideal x = \ideal y$

where $\ideal x$ and $\ideal y$ denote the ideals generated by $x$ and $y$ respectively.

Proof

We are to show that:

$x \divides y \text{ and } y \divides x \iff \ideal x = \ideal y$

Thus:

\(\ds \)

\(\)

\(\ds x \divides y \text{ and } y \divides x\)

Definition 1 of Associate in Integral Domain

\(\ds \)

\(\leadstoandfrom\)

\(\ds \ideal y \subseteq \ideal x \text{ and } \ideal x \subseteq \ideal y\)

Element in Integral Domain is Divisor iff Principal Ideal is Superset

\(\ds \)

\(\leadstoandfrom\)

\(\ds \ideal x = \ideal y\)

Definition 2 of Set Equality

$\blacksquare$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 62.3$ Factorization in an integral domain: $\text{(iii)}$