DESCENDING CHAIN-CONDITIONS AND GRADED RINGS

The structure of a group graded ring R satisfying certain classical fi niteness conditions is described module the homogeneous part of the Ja cobson radical J(gr)(R). It is shown that R/J(gr)(R) is a finite direc t product of matrix rings over group crossed products over division ri ngs. In the more general case of a semigroup graded ring R the structu re of R module its Jacobson radical can be described in terms of finit ely many group graded subrings. These subrings are shown to inherit th e considered finiteness conditions of R. As an application we derive r esults that show when a graded ring is Artinian, semiprimary, or perfe ct.