For an $A$-module $T$ and $T_{0} \in \operatorname{Add}(T)$, when is $\operatorname{Ann}(T) = \operatorname{Ann}(T_{0})$?

Question

Let $A$ be any unitary ring, let $T$ be an $A$-module and $T_{0} \in \operatorname{Add}(T)$. Clearly, $\operatorname{Ann}(T) \subseteq \operatorname{Ann}(T_{0})$. As the title says, I'm wondering under which conditions the annihilators are equal. I think I have one such condition, coming from $1$-tilting modules, but I would also like to know if it occurs in other circumstances. Here's my proof for the $1$-tilting case.

Let $T$ be a $1$-tilting, then there exists an exact sequence $$0 \rightarrow A_{A} \xrightarrow{\phi} T_{0} \rightarrow T_{1} \rightarrow 0$$ where $T_{0}, T_{1} \in \operatorname{Add}(T)$ and $\phi$ is a left $\operatorname{Gen}(T)$-approximation.

Note

• $\operatorname{Gen}(T)$ is the subcategory of $\operatorname{Mod}(A)$ consisting of modules $X$ such that there is a surjection $T^{(I)} \rightarrow X \rightarrow 0$, for some set $I$
• Let $\mathcal{D}$ be an additive category and $\mathcal{C}$ a subcategory. Then a map $M \xrightarrow{f} N$ with $N \in \mathcal{C}$ is a left $\mathcal{C}$-approximation of $M$ if for every object $C \in \mathcal{C}$, every map $M \rightarrow C$ factors through $f$.

So, clearly $\operatorname{Ann}(T) \subseteq \operatorname{Ann}(T_{0})$.

Let's show the other direction. We construct a universal map $u: A_{A}^{(I)} \rightarrow T$ where $I = \operatorname{Hom}_{A}(A_{A}, T)$, which is clearly surjective. Since $T \in \operatorname{Gen}(T)$, the map $u$ factors through the approximation $\phi^{(I)}: A_{A}^{(I)} \rightarrow T_{0}^{(I)}$ via some map $u'$. Now, for all $t \in T$ we have $t = f(\alpha)$ for some $\alpha \in A_{A}^{(I)}$, and $f(\alpha) = u'\phi^{(I)}(\alpha)$. Take some element $a \in \operatorname{Ann}(T_{0})$, then $$ta = f(\alpha)a = u'\phi^{(I)}(\alpha)a = u'(\phi^{(I)}(\alpha)a) = u'(0) = 0$$ so $\operatorname{Ann}(T) = \operatorname{Ann}(T_{0})$ in this case.

First of all I would like to know if my arguments are valid, and that this is indeed true in this case. Second, I would also like to know of other circumstances when the annihilators of $T$ and $T_{0}$ are equal.