On the structure of parabolic subgroups

The main aim of this paper is to show that the Jacobson and Brown-McCoy radicals of rings graded by free groups are homogeneous. As an application we get some information on the structure of the Jacobson radical of monomial rings. In particular we give a positive answer to a question posed in [12]. We extend also a result of [13] on the Brown-McCoy radical of polynomial rings in non-commutative variables. Actually this and the question of [12] motivated our studies.