Associated elements

Let $R$ be an integral domain . Elements $a,b\in R$ are associates if there exists a unit $u\in R^\times$ such that $a=ub$.

Being associates is an equivalence relation capturing “the same divisor up to invertible scaling.” Uniqueness statements for gcds , prime elements , and factorizations are typically only up to associates.

Examples:

• In $\mathbb{Z}$, $2$ and $-2$ are associates (multiply by the unit $-1$).
• In $k[x]$ with $k$ a field, $f$ and $cf$ are associates for any nonzero scalar $c\in k$.
• In $\mathbb{Z}$, $2$ and $4$ are not associates since the only units are $\pm 1$.