K\"othe's Problem, Kurosch-Levitzki Problem and Graded Rings

TL;DR

This paper investigates how properties like nilpotency and nilness in the degree-zero component of graded rings influence the entire ring, extending classical theorems and solving longstanding problems in ring theory.

Contribution

It generalizes the Dubnov-Ivanov-Nagata-Higman theorem to graded rings and establishes new links between graded rings and classical problems like K"othe's and Kurosh-Levitzki's.

Findings

Nil (of bounded index) degree-zero component implies nil (bounded index) ring under certain conditions.

Nilpotent degree-zero component implies the entire ring is nilpotent.

Graded and -commutative rings provide positive solutions to K"othe's and Kurosh-Levitzki problems.

Abstract

Let $\mathfrak{R}$ be an associative ring graded by left cancellative monoid $\mathsf{S}$, and $e$ the neutral element of $\mathsf{S}$. We study the following problem: if $\mathfrak{R}_e$ is nil, then is $\mathfrak{R}$ nil/nilpotent? We have proved that if $\mathfrak{R}_e$ is nil (of bounded index) and $\mathsf{f}$- commutative, then $\mathfrak{R}$ is nil (of bounded index). Later, we have shown that $\mathfrak{R}_e$ being nilpotent implies $\mathfrak{R}$ is nilpotent. Consequently, we have exhibited a generalization of Dubnov-Ivanov-Nagata-Higman Theorem for the graded algebras case. Furthermore, we have exhibited relations between graded rings and the problems of K\"{o}the and Kurosh-Levitzki. We have proved that graded rings and $\mathsf{f}$-commutative rings provide positive solutions to these problems.