Let $R$ is a commutative ring have identity element, and $R$ have a unique maximal ideal. Let $a,b \in R$ such that $\langle a \rangle = \langle b \rangle$. Show that $$a=bu$$ for some $u\in R^*$.
- Anybody can ask a question
- Anybody can answer
- The best answers are voted up and rise to the top
If $a=0$, then $b=0$ and there's nothing to prove, so assume $a,b\neq 0$. Since $a\in(b)$, you can write $a=br$ and similarly $b=as$ for $r,s\in R$. Thus $a=br=ars$, so $(rs-1)a=0$. If $r$ is not a unit, then neither is $rs$. So $rs$ lies in the unique maximal ideal of $R$. This means that $rs-1$ is not in the unique maximal ideal, because then we would have that $1=rs-(rs-1)$ is in it. So $rs-1$ is a unit, which implies $a=0$, contrary to assumption.
More generally, let $R$ be a ring where zero-divisors lie in every max ideal. It's trivial if $a = 0$. Else $b = sa,\ a = rb = rsa,\:$ so $\,a(1-rs) = 0.\:$ So $1\!-\!rs$ is $\,0\,$ or a zero-divisor, so is in every max ideal, hence $rs$ is a unit, lying in no max ideal $M$ (else both $\:1\!-\!rs,\ rs \in M\:$ so their sum $\:1\in M)$.
Remark $ $ For further generalizations see D. Anderson et al. When are Associates Unit Multiples?