On graded nil good rings

TL;DR

This paper introduces the concept of graded nil-good rings, explores their properties, extensions, and relationships with other classes of graded rings, providing new insights into their structure and behavior.

Contribution

It defines graded nil-good rings, studies their extensions, and compares their class with nil-good rings, advancing understanding of graded ring properties.

Findings

Graded nil-good rings are introduced and characterized.

Extensions to graded group rings and matrix rings are analyzed.

The relationship between nil-good and graded nil-good rings is clarified.

Abstract

In this paper we introduce and study the notion of a graded nil-good ring which is graded by a group. We investigate extensions of graded nil-good rings to graded group rings, Further, we discuss graded matrix ring extensions and trivial extensions of graded nil-good rings. Furthermore, we show that the class of graded rings which are nil-good and the class of graded nil-good rings are not comparable. Moreover, we discuss the question of when the nil-good property of the component, which corresponds to the identity element the grading group, implies that the whole graded ring is graded nil-good is also treated.