# Definition:Associate/Commutative and Unitary Ring

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## 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): Exercise $24.18$

- 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 6$: Rings and fields