# Associates are Unit Multiples

## Theorem

Let $\struct {D, +, \circ}$ be an integral domain whose unity is $1_D$.

Let $\struct {U_D, \circ}$ be the group of units of $\struct {D, +, \circ}$.

Then:

- $\forall x, y \in D: x \divides y \land y \divides x \iff \exists u \in U_D: y = u \circ x$

That is, if an element of $D$ is an associate of another element of $D$, then one is a unit multiple of the other.

## Proof

If $y = u \circ x$, then $x = u^{-1} y$, and by the definition of divisor, both $x \divides y$ and $y \divides x$.

Suppose $x \divides y$ and $y \divides x$.

Then:

- $\exists s, t \in D: y = t \circ x, x = s \circ y$

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 above | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle s \circ \paren {t \circ x}\) | $\quad$ from above | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \paren {s \circ t} \circ x\) | $\quad$ Definition of Associative Operation | $\quad$ |

So $s \circ t = 1_D$ and both $s \in U_D, t \in U_D$.

The result follows.

$\blacksquare$

## Sources

- 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 62.1 \ \text{(iv)}$