Definition:Associate/Integral Domain/Definition 3

Definition

Let $\struct {D, +, \circ}$ be an integral domain.

Let $x, y \in D$.

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

$y = u \circ x$

and consequently:

$x = u^{-1} \circ y$

That is, if and only if $x$ and $y$ are unit multiples of each other.

Also see

• Equivalence of Definitions of Associate in Integral Domain

• Results about associates can be found here.

Sources

• 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 6$: Rings and fields
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 62$. Factorization in an integral domain: $(2)$