We introduce a new large class of semigroupsSincluding all locally finite, completely regular and strongly π-regular linear semigroups. For any semigroupSin the class and anyS-graded ringR, the structure of the Jacobson radical ofRis reduced to the radicals of subrings graded by the maximal subgroups ofS. Many results on radicals follow from this reduction in a unified way. In two special cases the reduction is simplified.