The generalized zero-divisor conjecture for graded rings

Let $R=\bigoplus_{n i G}R_n$ be a ring graded by a torsion-free group $G$, and let $I$ be a \left ideal of $R$. The ideal $I$ is unfaithful when its annihilator is nonzero. The generalized zero-divisor conjecture. If $I$ is unfaithful, then there \exists a nonzero homogeneous element $g i R$ such that $gI=0$. The paper proves this assertion when $G$ is a totally ordered group, while the stated torsion-free-group version is not resolved in the supplied text.