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.
Best Answer
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.