Corollary 1

Let M be a nonzero finitely generated A-module. There exists a two-sided ideal $\mathfrak{a}$ of A, annihilator of a simple A-module, such that $\mathfrak{a}M$ is not equal to M.

Let N be a maximal submodule of M (Proposition 3), and let $\mathfrak{a}$ be the annihilator of the simple A-module $M/N$; it is a two-sided ideal of A, and we have $\mathfrak{a}(M/N) = 0$, whence $\mathfrak{a}M\subset N$ and consequently $\mathfrak{a}M\not= M$.