More on quasi-Frobenius rings

Let $R$ be a ring and $J$ its Jacobson radical. Let us set $J^1=J$, $J^\alpha=JJ^{\alpha-1}$, and $J^\alpha=\bigcap_{\beta<\alpha}J^\beta$ if $\alpha$ is a limit ordinal. We call a ring an annihilating ring if the left (right) annihilator of the right (left) annihilator of an arbitrary left (right) ideal $I$ is $I$ itself. We prove that a ring $R$ is quasi-Frobenius if and only if it is a left self-injective annihilating ring and $J^\alpha=0$ for some transfinite $\alpha$.
Bibliography: 15 titles.