DESCENDING CHAIN-CONDITIONS AND GRADED RINGS

The structure of a group graded ring R satisfying certain classical finiteness conditions is described module the homogeneous part of the Jacobson radical J(gr)(R). It is shown that R/J(gr)(R) is a finite direct product of matrix rings over group crossed products over division rings. In the more general case of a semigroup graded ring R the structure of R module its Jacobson radical can be described in terms of finitely many group graded subrings. These subrings are shown to inherit the considered finiteness conditions of R. As an application we derive results that show when a graded ring is Artinian, semiprimary, or perfect.