Results 1 to 2 of 2

Math Help - [SOLVED] maximal left ideal or not?

  1. #1
    Junior Member
    Joined
    Jan 2010
    Posts
    59
    Thanks
    3

    [SOLVED] maximal left ideal or not?

    Let R = Mn(F) be the ring consists of all n*n matrices over a field F and
    E = E11 + E22 + ... + En-1,n-1, where Eii is the elementary matrix (Eij is matrix whose ij th element is 1 and the others are 0) .
    I know that RE is a maximal left ideal . Let Q be an invertible matrix . Can we say that REQ is a maximal left ideal ?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by xixi View Post
    Let R = Mn(F) be the ring consists of all n*n matrices over a field F and
    E = E11 + E22 + ... + En-1,n-1, where Eii is the elementary matrix (Eij is matrix whose ij th element is 1 and the others are 0) .
    I know that RE is a maximal left ideal . Let Q be an invertible matrix . Can we say that REQ is a maximal left ideal ?
    yes. if REQ \subseteq I \lhd_{\ell} R, then RE \subseteq IQ^{-1}. now IQ^{-1} is a left ideal of R and so, since RE is a maximal left ideal, we must have either IQ^{-1}=R or IQ^{-1}=RE.

    thus either I=RQ=R or I=REQ.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. prove N is a maximal ideal iff N is a prime ideal
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: March 20th 2011, 09:02 AM
  2. [SOLVED] looking for a maximal left ideal
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: June 27th 2010, 03:58 PM
  3. maximal left ideal of matrix ring
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: March 3rd 2010, 07:21 AM
  4. Prime ideal or maximal ideal
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: January 21st 2010, 05:42 AM
  5. Maximal Ideal, Prime Ideal
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: September 28th 2008, 02:39 PM

Search Tags


/mathhelpforum @mathhelpforum