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Math Help - a question on a finite local ring

  1. #1
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    a question on a finite local ring

    Let R be a commutative finite local ring which is not a field . Let M be the maximal ideal of R . Can we conclude that  M\neq M^2 ?
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  2. #2
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    Quote Originally Posted by xixi View Post
    Let R be a commutative finite local ring which is not a field . Let M be the maximal ideal of R . Can we conclude that  M\neq M^2 ?
    yes. M is a finitely generated R module. so, by Nakayama's lemma, if IM=M with the ideal I \subseteq J(R)=M, then M=\{0\}, i.e. R must be a field. contradiction!
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