Let $A$ be an Artin algebra and $\mathscr{P}(A)$ the category of projective $A$-modules.
I don't know how to show the following facts:
All objects in $\mathscr{P}(A)$ are injective as objects of $\mathscr{P}(A)$ iff for each simple right $A$-module $S$ we have that $A$ contains a right $A$-submodule isomorphic to $S$.
mod-$A$ has no module of projective dimension 1 iff all objects of $\mathscr{P}(A)$ are injective as objects of $\mathscr{P}(A)$.
Thanks for the help.