ON HOMOGENEOUS STRONGLY α-n-IRREDUCIBLE IDEALS OF COMMUTATIVE GRADED RINGS
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Let M be a commutative additive monoid with identity element denoted by 0 and R be a commutative graded ring, which is graded by M. Let I be a homogeneous ideal of R, n be a positive integer, and α : ℒ(R) → ℒ(R) ∪ {∅} be a function, where ℒ(R) is the set of all homogeneous ideals of R. In this paper, we generalize the notion of strongly α-n-irreducible ideals to the context of graded rings, that is, if whenever I₁ ∩ ⋯ ∩ Iₙ₊₁ ⊆ I and I₁ ∩ ⋯ ∩ Iₙ₊₁ ⊈ α(I) for homogeneous ideals I₁, ..., Iₙ₊₁ of R, there are n of the Iᵢ’s whose intersection is in I. We study the transfer of this new concept in the idealization of graded modules and the amalgamation of graded rings along a homogeneous ideal.
Abstract
Let M be a commutative additive monoid with identity element denoted by 0 and R be a commutative graded ring, which is graded by M. Let I be a homogeneous ideal of R, n be a positive integer, and α : ℒ(R) → ℒ(R) ∪ {∅} be a function, where ℒ(R) is the set of all homogeneous ideals of R. In this paper, we generalize the notion of strongly α-n-irreducible ideals to the context of graded rings, that is, if whenever I₁ ∩ ⋯ ∩ Iₙ₊₁ ⊆ I and I₁ ∩ ⋯ ∩ Iₙ₊₁ ⊈ α(I) for homogeneous ideals I₁, ..., Iₙ₊₁ of R, there are n of the Iᵢ’s whose intersection is in I. We study the transfer of this new concept in the idealization of graded modules and the amalgamation of graded rings along a homogeneous ideal.
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