Suppose that $R$ is a finite dimensional $k$-algebra. I say that $R$ is Frobenius if it is locally bounded (see this question for a definition) and indecomposable projectives and injectives coincide. Could you help me to prove the following:
$R$ is Frobenius if and only if $_RR$ is injective as an $R$-module.
Of course if $R$ is Frobenius then $_RR$ is injective as an $R$-module, how can I prove the converse?