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

  • Thread starter rachellcb
  • Start date
  • #1
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Homework Statement


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?
 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
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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).
 

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