Lemma 15.15.2. An auto-associated ring $R$ has the following property: (P) Every proper finitely generated ideal $I \subset R$ has a nonzero annihilator.

Proof. By assumption there exists a nonzero element $x \in R$ such that for every $f \in \mathfrak m$ we have $f^ n x = 0$. Say $I = (f_1, \ldots , f_ r)$. Then $x$ is in the kernel of $R \to \bigoplus R_{f_ i}$. Hence we see that there exists a nonzero $y \in R$ such that $f_ i y = 0$ for all $i$, see Algebra, Lemma 10.24.4. As $y \in \text{Ann}_ R(I)$ we win. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).