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

Abstract

In this article, 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 if $I$ is an unfaithful left ideal of a $G$-graded ring $R$ with $G$ a totally ordered group, then there exists a (nonzero) homogeneous element $g\in R$ such that $gI=0$. This theorem gives an affirmative answer to the new conjecture in the case that the group involved in the grading is a totally ordered group. Our result also generalizes McCoy's famous theorem on polynomial rings to the more general setting of $G$-graded rings. Then we focus on Kaplansky's unit conjecture. Although this conjecture was recently disproved by a counterexample in the general case, we discovered quite useful and general results that give an affirmative answer to the generalized version of the unit conjecture in the case that the group involved in the grading is a totally ordered group. Especially, we show that every invertible element of a $G$-graded domain with $G$ a totally ordered group is homogeneous. This key result enables us to provide a characterization of invertible elements in $G$-graded commutative rings. This theorem, in particular, tells us that the homogeneous components of an invertible element form a co-maximal ideal and all distinct double products are nilpotent. Next, we prove that if $I$ is a graded radical ideal of a $G$-graded commutative ring $R$ with $G$ a torsion-free Abelian group and $J$ an arbitrary ideal of $R$, then the colon ideal $I:_{R}J$ is a graded ideal. Our theorem vastly generalizes Armendariz' result on reduced polynomial rings to the more general setting of graded rings.