Group Gradings on Associative Algebras

Let R = g G Rg be a G-graded ring. We describe all types of gradings on R if G is torsion free and R is Artinian semisimple. If R is a matrix algebra over an algebraically closed field F, then we give a description of all G-gradings on R provided that G is an abelian group. In the case of an abelian group G we also classify all finite-dimensional graded simple algebras and finite-dimensional graded division algebras over an algebraically closed field of characteristic zero.