Group graded rings with the nil-good property

In this article, we introduce and study the notion of graded nil-good ring graduated by a group. We prove that nil-good property of the component corresponding to the identity element of the grading group does not imply the graded nil-good property of the whole graded ring. We discuss the garded nil-good property in trivial ring extensions and graded group rings. We establish some sufficient conditions for a graded group ring to be graded nil-good. Furthermore, we give a sufficient condition for the graded matrix ring over a graded commutative nil-good ring to be graded nil-good.