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Ring problem: nilpotent elements and units

  1. Mar 18, 2012 #1
    1. The problem statement, all variables and given/known data
    Let R be a ring with multiplicative identity. Let u [itex]\in[/itex] R be a unit and let a1, ..., ak be nilpotent elements that commute with each other and with u. To show: u + a1 + ... + ak is a unit.

    3. The attempt at a solution
    Need to show that u'(u + a1 + ... + ak)=1 for some u' [itex]\in[/itex] R.
    I know that 1 - ai is a unit (i=1,2,...,k) since each ai is nilpotent.
    So i'm trying to think of possible candidates for u'.
    My attempt so far have been to try and cancel the ai terms by multiplying
    u + a1 + ... + ak by a1n-1, where a1n =0 and so on for all all a_i but this leaves me with ua1n-1...akm-1 and doesn't seem to be leading anywhere...
    Any suggestions?
     
  2. jcsd
  3. Mar 18, 2012 #2

    jgens

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    Gold Member

    Show first that the sum of two nilpotent elements which commute with each other is nilpotent (Hint: Generalize the binomial theorem to an arbitrary ring). Then use induction to show that the sum of k nilpotent elements which commute with each other is nilpotent. Then lastly prove that the sum of a nilpotent element and a unit is a unit (Hint: This is similar to proving 1+x is a unit, where x is nilpotent).
     
  4. Mar 20, 2012 #3
    Solved, thanks! :smile:
     
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