Let $R$ be a commutative ring with unit. Show that $\langle a\rangle = \langle b\rangle \iff a$ and $b$ are friends

Question

Let $R$ be a commutative ring with unit. $\langle a\rangle = \langle b\rangle \iff a$ and $b$ are friends.

So the direction of $a,b$ friends $\implies$ is trivial.

I can't manage to prove the other direction though. Can anyone help me?

I started with that: The ideals are equal so $a\in \langle b\rangle$ so there exists $c\in R$ such that $a=bc$. The same goes for exists $d\in R$ such that $b=ad$ so we get $a(1-dc)=0$ so I need to prove $1-dc$ isn't a zero divisor but got no idea how.

Friends means there exists $u\in R^{\times}$ such that $a=bu$ — Saar, Jan 17 '18 at 10:26
Oh really? In Hebrew it's called "haverim" which translates to "friends" xD — Saar, Jan 18 '18 at 23:34

score 2 · Answer 1 · answered Jan 17 '18 at 15:12

2

I can't manage to prove the other direction though.

It's not your fault, it is not possible to prove because there are counterexamples. (See also the linked questions to that question.)

The only reasonable context in which the converse you suggest is true is when $R$ is a domain, that is, cancellation is possible.

answered Jan 17 '18 at 15:12

rschwieb

153,510

Thanks you very much! I thought that might be the problem but you know... I usually think the TA knows everything better xD – Saar Jan 18 '18 at 23:37

Let $R$ be a commutative ring with unit. Show that $\langle a\rangle = \langle b\rangle \iff a$ and $b$ are friends

1 Answers1