# Units in Rings: show 1-ab a unit <=> 1-ba a unit

## Homework Statement

Let R be a ring with multiplicative identity. Let a, b $\in R$.
To show: 1-ab is a unit iff 1-ba is a unit.

## The Attempt at a Solution

Assume 1-ab is a unit. Then $\exists u\in R$ a unit such that (1-ab)u=u(1-ab)=1

$\Leftrightarrow$ u-abu=u-uab $\Leftrightarrow$ abu=uab. Not sure if this is useful, I haven't been able to go anywhere with it...

I also tried (1-ab)(1-ba)=1-ab-ba+abba but this isn't giving me any inspiration either.

Ideas on how to solve this?