Group graded rings with the nil-good property

Abstract

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.

Keywords:

• Graded rings and modules
• nil-good rings
• group rings
• matrix rings
• trivial ring extensions

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 16W50
• 16S34
• 16S50

Acknowledgment

The authors would like to express their sincere thanks for the referee for his/her helpful suggestions and comments.