Lemma 2

Let M be an A-module and N, P, and Q submodules of M. Suppose that we have $P\subset Q$, $N\cap P = N\cap Q$, and N + P = N + Q. We then have P = Q.

Let $x$ be an element of Q. It belongs to N + P; hence, there exists an element $y$ of P such that $x-y\in N$. Since Q contains P, the difference $x-y$ belongs to $N\cap Q$ and therefore to P. Consequently, $x$ belongs to P.