ON (n, 1)-ABSORBING δ-PRIMARY IDEALS
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Let R be a commutative ring with nonzero identity. Let I(R) be the set of all ideals of R and let δ : I(R) → I(R) be a function. Then δ is called an expansion function of ideals of R if whenever L, I, J are ideals of R with J ⊆ I, we have L ⊆ δ(L) and δ(J) ⊆ δ(I). Let δ be an expansion function of ideals of R. In this paper, we introduce and investigate a new class of ideals that is closely related to the class of δ-primary ideals. A proper ideal I of R is said to be an (n, 1)-absorbing δ-primary ideal if whenever nonunit elements a₁, a₂, ..., aₙ₊₁ ∈ R and a₁a₂...aₙ₊₁ ∈ I, then a₁a₂...aₙ ∈ I or aₙ₊₁ ∈ δ(I). Moreover, we give some basic properties of this class of ideals and we study the (n, 1)-absorbing δ-primary ideals of the localization of rings, the direct product of rings and the trivial ring extensions.
Abstract
Let R be a commutative ring with nonzero identity. Let I(R) be the set of all ideals of R and let δ : I(R) → I(R) be a function. Then δ is called an expansion function of ideals of R if whenever L, I, J are ideals of R with J ⊆ I, we have L ⊆ δ(L) and δ(J) ⊆ δ(I). Let δ be an expansion function of ideals of R. In this paper, we introduce and investigate a new class of ideals that is closely related to the class of δ-primary ideals. A proper ideal I of R is said to be an (n, 1)-absorbing δ-primary ideal if whenever nonunit elements a₁, a₂, ..., aₙ₊₁ ∈ R and a₁a₂...aₙ₊₁ ∈ I, then a₁a₂...aₙ ∈ I or aₙ₊₁ ∈ δ(I). Moreover, we give some basic properties of this class of ideals and we study the (n, 1)-absorbing δ-primary ideals of the localization of rings, the direct product of rings and the trivial ring extensions.
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