If $A \cong A^*$, is every projective module also injective?

Question

Suppose $A$ is a finite-dimensional algebra over $k$. Assume further that $A \cong A^* = \text{Hom}(A,k)$ as $A$-modules.

My question is: is every finite dimensional projective module over $A$ also injective?

Thanks for help!

Answer 1

$A^*$ is injective over $A$. Hence $A$ is injective over itself. But the finite dimensionality of $A$ implies that $A$ is Noetherian, so it's quasi-Frobenius (in fact it is even Frobenius).

Now if $P$ is projective over $A$, then $P \oplus Q = A^{\oplus m}$ for some $A$-module $Q$, but in a Noetherian ring direct summands of injectives are injective. Hence $P$ is injective.