Rings over which all modules are $I_0$-modules

Let $A$ be a ring that does not contain an infinite set of idempotents that are orthogonal modulo the ideal $\operatorname{SI}(A_A)$. It is proved that all $A$-modules are $I_0$-modules if and only if either $A$ is a right semi-Artinian right V-ring or $A/\operatorname{SI}(A_A)$ is an Artinian serial ring and the square of the Jacobson radical of $A/\operatorname{SI}(A_A)$ is equal to zero.