Definition:Associate/Commutative and Unitary Ring

Definition

Let $\struct {R, +, \circ}$ be a commutative ring with unity.

Let $x, y \in R$.

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

Also known as

The statement $x$ is an associate of $y$ can be expressed as $x$ is associated to $y$.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $24$. The Division Algorithm: Exercise $24.18$