Nakayama's Lemma for nilpotent Ideals

Let \(A\) be a ring, \(I\) a nilpotent ideal and \(M\) an \(A\)-module and \(N,N' \subseteq M\) submodules, such that

\begin{align*} M = N + I N' \end{align*}

Then it follows, that

\begin{align*} M = N \end{align*}

By Assuption there exists an \(n \in \mathbb{N}\), such that \(I^n = 0\). Since \(N' \subseteq M = N + IN'\), we get

\begin{align*} M =& N + IN' \\ \subseteq& N + I(N + IN') \\ =& N + IN + IIN' \end{align*}

Therfore, since \(N + IN = N\), we get

\begin{align*} (N + IN) + IIN' =& N + IIN' \\ \end{align*}

by repeating sufficiently often, we get

\begin{align*} N + \left(I^n\right)N' =& N + 0 \cdot N' \\ =& N \end{align*}