Homogeneity of zero-divisors, units and idempotents in a graded ring

TL;DR

This paper investigates the structure of zero-divisors, units, and idempotents in graded rings, extending classical conjectures to broader classes like monoid-rings and G-graded rings, with new generalizations and proofs.

Contribution

It generalizes Kaplansky's zero-divisor conjecture to G-graded rings and provides new results on the homogeneity of key elements in these rings.

Findings

Generalization of Kaplansky's zero-divisor conjecture to G-graded rings

Proven homogeneity properties of zero-divisors, units, and idempotents in graded rings

Extended classical results to monoid-rings and broader graded structures

Abstract

In this article we prove several important results on graded rings, especially monoid-rings, that are motivated and inspired by Kaplansky's zero-divisor, unit and idempotents conjectures. Among the main results, we first generalize Kaplansky's zero-divisor conjecture of group-rings $K[G]$ (with $K$ a field) to the more general setting of $G$-graded rings $R=\bigoplus\limits_{n\in G}R_{n}$ with $G$ a torsion-free group. Then we prove that ...