# Ring theory associates

Can someone please give me an example of of this definition, as I am finding it hard to get my head around or even understand what an "associate" is.

Let $R$ be a commutative ring with unity. Elements $a$ and $b$ are called associates in $R$ if $b =ua$ for some unit $u$ of $R$.

My notes don't provide an example. Can someone please explain it further or even better show me an example of this definition?

Much appreciated.

Thank you.

In the ring of integers $\mathbb{Z}$, the units are precisely the integers $1$ and $-1$. If $a,b\in\mathbb{Z}$ are associates, then by definition $a=ub$ for some unit $u$, so that either $a=1\cdot b=b$ or $a=(-1)\cdot b=-b$. Thus, two integers are associates in $\mathbb{Z}$ if and only if they're the same up to sign.