Communications of the
Korean Mathematical Society

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024



Commun. Korean Math. Soc. 2023; 38(4): 1001-1017

Online first article October 16, 2023      Printed October 31, 2023

Copyright © The Korean Mathematical Society.

On graded $N$-irreducible ideals of commutative graded rings

Anass Assarrar, Najib Mahdou

Box 2202, University S.M. Ben Abdellah Fez; Box 2202, University S.M. Ben Abdellah Fez


Let $R$ be a commutative graded ring with nonzero identity and $n$ a positive integer. Our principal aim in this paper is to introduce and study the notions of graded $n$-irreducible and strongly graded $n$-irreducible ideals which are generalizations of $n$-irreducible and strongly $n$-irreducible ideals to the context of graded rings, respectively. A proper graded ideal $I$ of $R$ is called graded $n$-irreducible (respectively, strongly graded $n$-irreducible) if for each graded ideals $I_{1}, \ldots,I_{n+1}$ of $R$, $I=I_{1} \cap \cdots \cap I_{n+1}$ (respectively, $I_{1} \cap \cdots \cap I_{n+1} \subseteq I$ ) implies that there are $n$ of the $I_{i}$ 's whose intersection is $I$ (respectively, whose intersection is in $I$). In order to give a graded study to this notions, we give the graded version of several other results, some of them are well known. Finally, as a special result, we give an example of a graded $n$-irreducible ideal which is not an $n$-irreducible ideal and an example of a graded ideal which is graded $n$-irreducible, but not graded $(n-1)$-irreducible.

Keywords: $n$-irreducible ideals, graded $n$-irreducible ideals, strongly graded $n$-irreducible ideals, graded $n$-absorbing ideals

MSC numbers: 13A02, 13A15

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