$E$-solvable modules

$E$-nilpotent and $E$-solvable modules have been defined. Some properties of such modules have been proved. For instance, all direct summands of an $E$-nilpotent module are fully invariant, and the $E$-commutant of an $E$-solvable module is contained in the intersection of all maximal commutatorically invariant submodules. Necessary and sufficient conditions under which a finite length module is $E$-solvable have been found.