Equivalence of Definitions of Associates

Theorem
Let $\left({D, +, \circ}\right)$ be an integral domain.

Let $a, b \in D$.

Then:
 * $a \backslash b, b \backslash a$

iff:
 * $a = u b$ where $u$ is a unit of $D$.

That is, the two definitions of $a$ being an associate of $b$ are logically equivalent.

Proof

 * Let $a \backslash b, b \backslash a$.

Then $a = c b, b d = a$.

So $a c d = a$ and so $c d = 1$, as $a$ is cancellable in an integral domain.

So $d$ is a unit of $D$.


 * Suppose $a = u b$ where $u$ is a unit of $D$.

Thus $b \backslash a$.

Then $u^{-1} a = b$ where $u^{-1} \in D$.

So by definition $a \backslash b$.