# Equivalence of Definitions of Associate in Integral Domain/Definition 1 Equivalent to Definition 3

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## 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 3

$x$ and $y$ are associates (in $D$) if and only if there exists a unit $u$ of $\struct {D, +, \circ}$ such that:

$y = u \circ x$

and consequently:

$x = u^{-1} \circ y$

That is, if and only if $x$ and $y$ are unit multiples of each other.

## Proof

 $\displaystyle y$ $=$ $\displaystyle u \circ x$ $\quad$ $\quad$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle x$ $=$ $\displaystyle u^{-1} \circ y$ $\quad$ Definition of Unit of Ring $\quad$

By the definition of divisor:

$x \divides y$ and $y \divides x$

$\Box$

Let $x \divides y$ and $y \divides x$.

Then $\exists s, t \in D$ such that:

$(1): \quad y = t \circ x$

and:

$(2): \quad x = s \circ y$

If either $x = 0_D$ or $y = 0_D$, then so must be the other (as an integral domain has no zero divisors by definition).

So $x = 1_D \circ y$ and $y = 1_D \circ x$, and the result holds.

Otherwise:

 $\displaystyle 1_D \circ x$ $=$ $\displaystyle x$ $\quad$ Definition of Unity of Ring $\quad$ $\displaystyle$ $=$ $\displaystyle s \circ y$ $\quad$ from $(2)$ $\quad$ $\displaystyle$ $=$ $\displaystyle s \circ \paren {t \circ x}$ $\quad$ from $(1)$ $\quad$ $\displaystyle$ $=$ $\displaystyle \paren {s \circ t} \circ x$ $\quad$ Definition of Associative Operation $\quad$

So:

$s \circ t = 1_D$

and both $s \in U_D$ and $t \in U_D$.

The result follows.

$\blacksquare$