SOLVED a question on a finite local ring

Jan 2010
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Let R be a commutative finite local ring which is not a field . Let \(\displaystyle M\) be the maximal ideal of R . Can we conclude that \(\displaystyle M\neq M^2\) ?
 

NonCommAlg

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