Results 1 to 5 of 5

Math Help - Ring with 0 only nilpotent element

  1. #1
    Newbie
    Joined
    Oct 2008
    Posts
    22

    Ring with 0 only nilpotent element

    Let A be a ring in wich 0 is the only nilpotent element ( x^n=0 \rightarrow x=0  ) . Prove that x is invertible if and only if x is left invertible .
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by petter View Post
    Let A be a ring in which 0 is the only nilpotent element ( x^n=0 \rightarrow x=0  ) . Prove that x is invertible if and only if x is left invertible .
    Hint: Let a be a left inverse for x. Then x(1-xa) is nilpotent (just check that its square is 0).
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Oct 2008
    Posts
    22
    Quote Originally Posted by Opalg View Post
    Hint: Let a be a left inverse for x. Then x(1-xa) is nilpotent (just check that its square is 0).
    So x(1-xa)=0 . How to continue ?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by petter View Post
    So x(1-xa)=0 . How to continue ?
    Multiply on the left by a.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Oct 2008
    Posts
    22
    Quote Originally Posted by Opalg View Post
    Multiply on the left by a.
    I'm stupid .
    Thx !
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: November 30th 2011, 08:10 PM
  2. ideal,nil,nilpotent ideal in prime ring
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 24th 2011, 08:57 AM
  3. Nilpotent Element of a Ring
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: July 23rd 2010, 02:15 PM
  4. Replies: 3
    Last Post: March 23rd 2010, 08:05 PM
  5. Element of a Group Ring
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: May 24th 2009, 12:50 PM

Search Tags


/mathhelpforum @mathhelpforum