# Thread: a question on a finite local ring

1. ## 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 $M\neq M^2$ ?

2. Originally Posted by xixi
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!