# G-graded irreducibility and the index of reducibility

## TL;DR

This paper generalizes the concept of irreducibility to G-graded modules over Noetherian rings and explores the relationship between graded and usual irreducibility, providing new inequalities for the index of reducibility.

## Contribution

It introduces G-graded irreducibility, proves its equivalence to usual irreducibility, and establishes inequalities for indices of reducibility in the graded setting.

## Key findings

- G-graded irreducibility is equivalent to usual irreducibility.
- Established inequalities for indices of reducibility between radical non-graded ideals and their graded subideals.
- Generalized previous results from the -graded case to arbitrary torsionfree abelian groups.

## Abstract

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.