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

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 $R=\underset{n\in G}{⨁}{R}_n$ with $G$ a torsion-free group. Then we prove that …