Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: G-graded irreducibility and the index of reducibility   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.

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The categories of this paper are: atom-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: plasm-ph
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The categories of this paper are: math-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: ao-sci
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The categories of this paper are: q-alg
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The categories of this paper are: dg-ga
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The categories of this paper are: solv-int
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The categories of this paper are: funct-an
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The categories of this paper are: physics
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The categories of this paper are: supr-con
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The categories of this paper are: mtrl-th
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The categories of this paper are: bayes-an
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The categories of this paper are: gr-qc
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The categories of this paper are: chao-dyn