  Necessary and sufficient conditions are given for a $G$-graded simple module
over a unital associative algebra, graded by an abelian group $G$, to be
isomorphic to a loop module of a simple module, as well as for two such loop
modules to be isomorphic to each other. Under some restrictions, these loop
modules are completely reducible (as ungraded modules), and some of their
invariants --- inertia group, graded Brauer invariant and Schur index --- which
were previously defined for simple modules over graded finite-dimensional
semisimple Lie algebras over an algebraically closed field of characteristic
zero, are now considered in a more general and natural setting.
