I'm wondering if $1$-tilting modules (tilting modules of projective dimension $\leq 1$) are always faithful, as it seems to me that they should be. Here's my attempted proof, I'd love some feedback, most importantly if I have made errors.
The modules will be right modules.
Let $A$ be some unary ring, and $T \in \operatorname{Mod}(A)$ a tilting module. Then there is an exact sequence
\begin{equation*}
0 \rightarrow A_{A} \xrightarrow{\phi} T_{0} \rightarrow T_{1} \rightarrow 0
\end{equation*}
where $T_{0}, T_{1}$ lie in $\operatorname{Add}(T)$ and $\phi$ is a left $\operatorname{Gen}(T)$-approximation.
$\operatorname{Gen}(T)$ is the category of $A$-modules $X$ such that there exists a surjection $T^{(n)} \rightarrow X \rightarrow 0$.
Now, let $a \in \operatorname{Ann}(T)$. Since $T_{0} \oplus T' \cong T^{(I)}$ for some $A$-module $T'$ and some set $I$, we have that $T_{0}\cdot a = 0$.
Then $\phi(a) = \phi(1)\cdot a = 0$, so $a \in \operatorname{Ker}(\phi) = (0)$, i.e. $\operatorname{Ann}(T) = (0)$.
Best Answer
Indeed every tilting module is faithful. Note that a general module $M$ is faithful if and only if there is an embedding of the algebra $A$ into $M^n$ for some $n$, see for example Lemma 5.5. in the book Frobenius algebras I by Skowronski and Yamagata.