DESCENDING CHAIN-CONDITIONS FOR GRADED RINGS

It is proved that if R is a perfect (resp. Artinian) strongly graded ring whose ground subring is, modulo its Jacobson radical, a finite direct product of finite-dimensional simple algebras over (nondenumerable) algebraically closed fields, then the grading group cannot contain an infinite abelian subgroup (resp. must be finite). These results extend those of A. Reid and D. S. Passman on twisted group algebras.