Let $R$ be a ring, $J(R)$ its Jacobson radical, $M$ is a finite $R$-module. The following statement is usually called Nakayama's lemma: if $IM=M$ then $M=0$. This is true over any ring (commutative or not).
Over a commutative ring the generalized version of Nakayama's lemma holds: if $IM=M$ for some ideal $I$ and finite $M$ then there is $x \in I$ such that $(1-x)M=0$.
The only proof I know is based on determinant trick, which doesn't generalize to non-commutative situation. I doubt that this is true over non-commutative ring. What are the simplest examples of a non-commutative ring and finite module over it that generalized Nakayama's lemma is false?