# 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.

## 1 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.

• oh thank you, ! makes more sense now, I dont know why i found this hard to understand..... – italy Jun 4 '15 at 21:41
• Glad I could help! – Zev Chonoles Jun 4 '15 at 21:42
• If this answer is satisfactory, you should soon be able to click a green checkmark to "accept" it. – Zev Chonoles Jun 4 '15 at 21:49
• Come on, @ZevChonoles, don't beg for votes! Your reputation is already solid! – Alex M. Jun 4 '15 at 21:50
• @AlexM: There's also value in taking it off the list of questions without an accepted answer. – Zev Chonoles Jun 4 '15 at 21:52