Abstract
Let R be a commutative Noetherian ring graded by a torsion-free 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 -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.
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