$I_0^*$-modules

We study rings over which every module is an $I_0^*$-module dual to $I_0$-module. We describe semiregular rings over which every module is simultaneously $I_0^*$-module and $I_0$-module. We give a description of rings over which every module is a direct sum of injective module and $SV$-module. We investigate relations between weakly Baer modules and $I_0^*$-modules.