Let $k$ be a field and $A$ a finite dimensional $k$-algebra.
The map $n \mapsto f_n$ where $f_n:A \to N$ is given by $f_n(a)=n a$ defines an embedding of $N$ into a $(A^e,1\otimes A^{op})$-injective module, $N \to Hom_k(A,N)$, for all left $A^e$-module $N$. Where $A^e=A \otimes_k A^{op}$ is the enveloping algebra.
I have been trying to construct in a similar way, an explicit $(A^e,1\otimes A^{op})$-projective module $P(N)$, together with a surjection $P(N) \to A$ for every left $A^e$-module $N$, but have not been succesfull.
Any ideas?