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Math Help - Idempotents and Projective R-Modules

  1. #1
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    Idempotents and Projective R-Modules

    Let e in R be idempotent. Show that the ideal Re is a projective R module.

    I want to do this without explicitly lifting. But I am struggling to write Re + M = N such that N is free.
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  2. #2
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    Quote Originally Posted by robeuler View Post
    Let e in R be idempotent. Show that the ideal Re is a projective R module.

    I want to do this without explicitly lifting. But I am struggling to write Re + M = N such that N is free.
    Hint: Re \oplus R(1-e)=R.
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  3. #3
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    Re: Idempotents and Projective R-Modules

    Do this hint work? I think R(1-e) = R because 1-e is unit by (1-e)(1+e)=1 if e is idempotent.
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