Graded radicals of graded rings

Abstract

Let G be a group and A a radical property in the category of associative rings. Using the generalized smash product of [1], we introduce a method for defining a corresponding radical property Arcf in the category of associative G-graded rings and grade-preserving ring homomorphisms. We investigate the properties of these new radicals and compare them with graded radicals which have been previously studied. For A = J, the Jacobson radical, ArefiS the usual graded Jacobson radical. (See for example [2], [7].) If A is the prime radical, then for G finite and R a G-graded ring, Ara(R) is the graded prime radical of [3], i.e. the intersection of the graded prime ideals of R. However, this intersection of graded ideals may be properly contained in Aref(R) for G infinite. If A is the strongly prime radical, then )~ref is the graded strongly prime radical of [8] for G finite, but again may properly contain this ideal for G infinite. We also discuss the cases of A equal to the Levitzski, Brown-McCoy and yon Neumann regular radicals, and compare Ara to suitable intersections of graded ideals.