A note on Nil and Jacobson radicals in graded rings

TL;DR

This paper extends Bergman's result by proving that the nil radical of a Z-graded ring is homogeneous, and explores properties of subrings generated by homogeneous elements, with implications for graded-nil rings.

Contribution

It demonstrates that the nil radical in Z-graded rings is homogeneous and analyzes the behavior of subrings generated by homogeneous elements in graded Jacobson radical rings.

Findings

Nil radical of Z-graded rings is homogeneous.

Subrings generated by homogeneous elements in graded Jacobson radical rings are Jacobson radical.

A ring with all subrings Jacobson radical is nil.

Abstract

It was shown by Bergman that the Jacobson radical of a Z-graded ring is homogeneous. This paper shows that the analogous result holds for nil rings, namely, that the nil radical of a Z-graded ring is homogeneous. It is obvious that a subring of a nil ring is nil, but generally a subring of a Jacobson radical ring need not be a Jacobson radical ring. In this paper it is shown that every subring which is generated by homogeneous elements in a graded Jacobson radical ring is always a Jacobson radical ring. It is also observed that a ring whose all subrings are Jacobson radical rings is nil. Some new results on graded-nil rings are also obtained.