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

[rachellcb]

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?

 

[Dick]

Ok, so u(1-ab)=1. You want to get ba into the picture somehow. So multiply on the left by b and on the right by a. So bu(1-ab)a=ba. Now the big hint is that (1-ab)a=a(1-ba).