Group gradings on associative algebras

Let R = circle plus (g is an element ofG) R-g 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. (C) 2001 Academic Press.