Let $A$ be any ring and consider modules on the left. For $M$ $A$-module, the trace $Tr(M,A)$ is a two-sided ideal of $A$. If $A$ is a unitary ring then:

- $Tr(P,A)P=P$, for $P$ projective;
- $Tr(P,A)^2=Tr(P,A)$.

Further, if $A$ is semiprimary then $I$ is an idempotent ideal of $A$ if and only if $I=Tr(P,A)$, for some finitely generated projective $P$, if and only if $I=AeA$, for some idempotent $eāA$.

In parallel, it is easy to check that if $A$ is unitary then the reject $Rej(A,M)$ is the (left) annihilator of the module $M$.

**Is there a characterisation of the reject of injective modules that is "somehow" dual to the one above?**

I know that in sufficiently good contexts there is a connection between annihilators of simple modules and prime ideals... It would be nice to have something for the reject of injectives that would hold for A semiprimary (or artinian or, in the worst case scenario, an Artin algebra).

Just one more thing. In the paper "Homological Theory of Idempotent Ideals" (Auslander, Platzeck, Todorov) we can find different characterisations of $k$-idempotent ideals using $\operatorname{Ext}$. If the base ring has a self-duality, $k$-idempotent ideals may also be characterised in terms of $\operatorname{Tor}$. So...

We have that an ideal $I$ of $A$ is idempotent if and only if $\operatorname{Ext}_A^1(A/I,Y)=0$ for all $Y$ in $\operatorname{mod} A/I$ (there are many other similar characterisations).

I was trying to see if it is true that: $I=Rej(A,Q)$, for some injective $Q$, if and only $\operatorname{Tor}^A_1(A/I,Y)=0$ for all $Y$ in $\operatorname{mod} A/I$...

(Maybe the assertion "$I=Rej(A,Q)$, for some injective $Q$, if and only $\operatorname{Ext}_A^1(Y,I)=0$ for all $Y$ in $\operatorname{mod} A/I$" makes more sense.

The problem is that projectives and injectives are not quite "dual" in $\operatorname{Mod}A$. The proofs I know for the properties that relate traces and projectives use the dual basis lemma.)