Let R be a ring with multiplicative identity. Let a, b [itex]\in R[/itex].
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 [itex]\exists u\in R[/itex] a unit such that (1-ab)u=u(1-ab)=1
[itex]\Leftrightarrow[/itex] u-abu=u-uab [itex]\Leftrightarrow[/itex] 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?