Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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ON GRADED N-IRREDUCIBLE IDEALS OF COMMUTATIVE GRADED RINGS

  • Anass Assarrar (Modelling and Mathematical Structures Laboratory Department of Mathematics Faculty of Science and Technology of Fez Box 2202, University S.M. Ben Abdellah Fez Morocco) ;
  • Najib Mahdou (Modelling and Mathematical Structures Laboratory Department of Mathematics Faculty of Science and Technology of Fez Box 2202, University S.M. Ben Abdellah Fez Morocco)
  • Received : 2023.01.07
  • Accepted : 2023.04.04
  • Published : 2023.10.31
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Abstract

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 I1, . . . , In+1 of R, I = I1 ∩ · · · ∩ In+1 (respectively, I1 ∩ · · · ∩ In+1 ⊆ I ) implies that there are n of the Ii '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.

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References

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