  Let $R$ be a commutative Noetherian ring graded by a torsionfree abelian
group $G$. We introduce the notion of $G$-graded irreducibility and prove that
$G$-graded irreducibility is equivalent to irreducibility in the usual sense.
This is a generalization of Chen and Kim's result in the $\mathbb{Z}$-graded
case. We also discuss the concept of the index of reducibility and give an
inequality for the indices of reducibility between any radical non-graded ideal
and its largest graded subideal.
