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Math Help - Question in Advanced Algebra

  1. #1
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    Question in Advanced Algebra

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  2. #2
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    Re: Question in Advanced Algebra

    it suffices to exhibit just ONE counter-example, so let's pick an easy A to work with:

    let A = Z+Z, the direct sum of the additive group (Z,+), where Z is the integers we know and love.

    let f(m,n) = (n,0). first, i show that f is indeed in End(A):

    f((m,n)+(m',n')) = f((m+m',n+n')) = (n+n',0) = (n,0) + (n',0) = f(m,n) + f(m',n').

    next i show that f is nilpotent:

    f2((m,n)) = f(f((m,n))) = f((n,0)) = (0,0), so f2 is the zero-map.

    now, let g(m,n) = (0,m). prove that g is likewise a nilpotent endomorphism of Z+Z.

    next, show f+g is NOT nilpotent (where (f+g)(m,n) = f((m,n)) + g((m,n)).

    ********

    for the first problem, consider (1A - f)(1A + f + f2 +....+ fn-1).
    Last edited by Deveno; November 21st 2012 at 01:25 PM.
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    Re: Question in Advanced Algebra

    Thank you very much
    Excuse me please clarify the problem(1A - f)is an invertible endomorphism (has inverse)
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  4. #4
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    Re: Question in Advanced Algebra

    Have you done what Deveno suggested? What is (1A - f)(1A + f + f2 +....+ fn-1).
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    Re: Question in Advanced Algebra

    Answer the question
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