It is well known that if $\mbox{Ext}^1_{A}(P,A/I)=0$ for all $I,$ then $\mbox{Ext}^i_{A}(P,A/I)=0$ for all $i$ and for all $I $and $P$ is projective.
We can also characterize a projective module $P$ by his trace ideal denoted $t(P),$ we can then for all projective module $P$ the following relations: $$Pt(P)=P,\tag{i}$$ $$t(P)^2=t(P).\tag{ii}$$
