Associatehood is Equivalence Relation

Theorem
Let $\left({D, +, \circ}\right)$ be an integral domain whose zero is $0_D$ and whose unity is $1_D$.

Let $\sim$ be the relation defined on $D$ as:

$\forall x, y \in D: x \sim y$ iff $x$ is an associate of $y$

Then $\sim$ is an equivalence relation.

Proof
Checking in turn each of the critera for equivalence:

Reflexive
Clearly $x \backslash x$ as $x = 1_D \circ x$, so $x \sim x$.

Symmetric
By the definition, $x \sim y \iff x \backslash y \land y \backslash x \iff y \sim x$.