nakayama's lemma for noncommutative rings

Let \(A\) be a ring, \(M\) a finitely generated module and \(\mathfrak{a} \subseteq \mathfrak{J}\) a subset of the Jacobson radical such that

\begin{align*} \mathfrak{a}M = M \end{align*}

Then \(M = 0\)

Assume \(M \neq 0\), then there exists a maximal proper submodule (cf. finitely generated module contains maximal submodule)
Then \(M/N\) is a simple module (cf. correspondence theorem for quotient modules).
Hence it is isomorphic to some \(R/\mathfrak{m}\) for some left maximal ideal \(\mathfrak{m}\).

Then

\begin{align*} M =& \mathfrak{m} M \cong& \mathfrak{m} (R/\mathfrak{m}) \\ =& 0 \end{align*}

contradiction