Exercise 23, § 9

Let A be a ring and $\mathfrak{a}$ a (left) ideal of A. Suppose that for every sequence $(a_n)_{n\geqslant 1}$ of elements of $\mathfrak{a}$, there exists an integer $m$ such that $a_1\cdots a_m= 0$. Prove that we have $\mathfrak{a}M\not= M$ for every nonzero A-module M (if not, construct, by induction, sequences $(a_n)_{n\geqslant 1}$ in $\mathfrak{a}$ and $(x_n)_{n\geqslant 1}$ in M such that we have $a_1\cdots a_px_p\not= 0$ for every $p$).