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

  1. Mar 21, 2012 #1
    1. The problem statement, all variables and given/known data
    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.

    3. 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?
     
  2. jcsd
  3. Mar 21, 2012 #2

    Dick

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    Science Advisor
    Homework Helper

    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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