let $A$ be an Artin algebra and $M$ in $\mathfrak{mod}\ A$.
Let $A$ be left-QF-3 with minimal faithful left ideal $Ae$. Then the following are equivalent:
$\bullet$ $Ae$-dom.dim.$(A)\geq 2$
$\bullet$ There holds a double centralizer property $A=\text{End}(Ae_{eAe})$
(This is theorem 2.10 of http://www.sciencedirect.com/science/article/pii/S002186930098726X#).
I'm having trouble with proving $\Rightarrow$.
I know that the double centralizer property $A=\text{End}(Ae_{eAe})$ holds if and only if there is an exact sequence $0\rightarrow A \stackrel{f}{\rightarrow} \underbrace{Ae\times \cdots\times Ae}_{n\ \text{times}} \rightarrow \underbrace{Ae\times \cdots\times Ae}_{m\ \text{times}}$
for some natural numbers $n$ and $m$, where $f$ is a left-$\mathfrak{add}\ Ae$-approximation, but I don't know, if that helps.
We have $Ae$-dom.dim.$(A)\geq 2$ and that only means that there is an exact sequence $0 \rightarrow A \rightarrow T_1 \rightarrow T_2$ with $T_1, T_2\in \mathfrak{add}\ {Ae}$. I would be happy, if I knew, how to prove that there is a sequence with the additional properties ($f$ minimal approximation and the targets of the maps are ${Ae}^n$ and ${Ae}^m$).
Thanks for the help!