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 $\displaystyle M\neq M^2$ ?
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!