Associates in rings that differ by more than sign

Question

Are there any rings where associates differ by more than just sign (e.g., 1 or -1)?

So far everything I've learned either involved associates differing by sign or a constant multiple, but I'm curious if there are other cases where the associates differ by more. I can't think of any...

Answer 1

For an example, take your favorite ring $R$ with $R^\times$ something else than $\{\pm 1\}$. For example $R=\mathbf{Z}[i]$, then $\mathbf{Z}[i]^\times=\{ \pm 1,\pm i\}$ so in there $3+2i$ and $2-3i$ are associate!

Answer 2

Elements $a,b\in R$ in an integral domain are called associated if we have $b=au$ for a unit $u\in R^{\times}$. So your question is equivalent to the question for which domains the units group is different from $\{\pm 1\}$.

Examples are the certain rings of integers in number fields, such as $$ \Bbb Z[\omega]^{\times}=\{\pm 1,\pm \omega,\pm \omega^2\} $$ where $\omega$ is a primitive third root of unity.

Reference: Determine all units in $\mathbb{Z}[\omega] := \{a+b\omega\mid a,b\in\mathbf{Z}\}$ where $\omega = \frac{-1 + i \sqrt{3}}{2}$