In this paper we introduce and study the notion of a graded (strongly) nil clean ring which is group graded. We also deal with extensions of graded (strongly) nil clean rings to graded matrix rings and to graded group rings. The question of when nil cleanness of the component, which corresponds to the neutral element of a group, implies graded nil cleanness of the whole graded ring is examined. Similar question is discussed in the case of groupoid graded rings as well.

In this paper we study the notion of a graded ?-group (X;+ ?), but graded in the sense of M. Krasner, i.e., we impose nothing on the grading set except that it is nonempty, since operations of and the grading of (X,+) induce operations (generally partial) on the grading set. We prove that graded ?-groups in Krasner?s sense are determined up to isomorphism by their homogeneous parts, which, with respect to induced operations, represent partial structures called ?-homogroupoids, thus narrowing down the theory of graded -groups to the theory of ?-homogroupoids. This approach already proved to be useful in questions regarding A. V. Kelarev?s S-graded rings inducing S; where S is a partial cancellative groupoid. Particularly, in this paper we prove that the homogeneous subring of a Jacobson S-graded ring inducing S is Jacobson under certain assumptions. We also discuss the theory of prime radicals for ?-homogroupoids thus extending results of A. V. Mikhalev, I. N. Balaba and S. A. Pikhtilkov in a natural way. We study some classes of ?-homogroupoids for which the lower and upper weakly solvable radicals coincide and also, study the question of the homogeneity of the prime radical of a graded ring.

In this paper, a graded ring is a ring which is the direct sum of a family of its additive subgroups indexed by a nonempty set under the assumption that the product of homogeneous elements is again homogeneous. We study graded special radicals and special radicals of graded rings, but which contain the corresponding Jacobson radicals. There are two versions of this graded radical, which we name the graded over-Jacobson and the large graded over-Jacobson radical. We establish several characterizations of the graded over-Jacobson radical of a graded ring and also prove that the largest homogeneous ideal contained in the corresponding classical radical of a graded ring coincides with the large graded over-Jacobson radical of that ring.

We investigate the graded Brown-McCoy and the classical Brown-McCoy radical of a graded ring, which is the direct sum of a family of its additive subgroups indexed by a nonempty set, under the assumption that the product of homogeneous elements is again homogeneous. There are two kinds of the graded Brown-McCoy radical, the graded Brown-McCoy and the large graded Brown-McCoy radical of a graded ring. Several characterizations of the graded Brown-McCoy radical are given, and it is proved that the large graded Brown-McCoy radical of a graded ring is the largest homogeneous ideal contained in the classical Brown-McCoy radical of that ring.

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