Homogeneity of idempotents and the Jacobson radical in a graded ring

Abstract: In this article, we first prove a key result which asserts that every nonzero idempotent element of a $G$-graded commutative ring with $G$ a torsion-free Abelian group is homogeneous of degree zero. This important special case leads us to derive a general result which asserts that every idempotent element of a $G$-graded commutative ring $R=\bigoplus\limits_{n\in G}R_{n}$ with $G$ an Abelian group is contained in the subring $\bigoplus\limits_{h\in H}R_{h}$ where $H$ is the torsion subgroup of $G$. This theorem gives an affirmative answer to the generalized version of Kaplansky's idempotent conjecture in the commutative case. Our result also ...