Trivial extensions satisfying certain valuation-like properties

Dobbs, David E.; Khalfi, Abdelhaq El; Mahdou, Najib

doi:10.1080/00927872.2018.1527926

Cited by 12 publications

(2 citation statements)

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“…Trivial ring extensions have been studied or generalized extensively, often because of their usefulness in constructing new classes of examples of rings satisfying various properties (cf. [1,6,7,11] ). In addition, for an ideal I of…”

Section: Resultsmentioning

confidence: 99%

On 1-absorbing δ-primary ideals

Khalfi

Mahdou

Teki̇r

et al. 2021

Analele Universitatii "Ovidius" Constanta - Seria Matematica

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Let R be a commutative ring with nonzero identity. Let 𝒥(R) be the set of all ideals of R and let δ : 𝒥 (R) → 𝒥 (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 a 1-absorbing δ -primary ideal if whenever nonunit elements a, b, c ∈ R and abc ∈ I, then ab ∈ I or c ∈ δ (I). Moreover, we give some basic properties of this class of ideals and we study the 1-absorbing δ-primary ideals of the localization of rings, the direct product of rings and the trivial ring extensions.

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Section: Resultsmentioning

confidence: 99%

On 1-absorbing δ-primary ideals

Khalfi

Mahdou

Teki̇r

et al. 2021

Analele Universitatii "Ovidius" Constanta - Seria Matematica

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Section: Is An Expansion Function Of I(r)mentioning

confidence: 99%

On $1$-absorbing $δ$-primary ideals

Khalfi,

Mahdou,

Tekir

et al. 2021

Preprint

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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 a 1-absorbing δ-primary ideal if whenever nonunit elements a, b, c ∈ R and abc ∈ I, then ab ∈ I or c ∈ δ(I). Moreover, we give some basic properties of this class of ideals and we study the 1-absorbing δ-primary ideals of the localization of rings, the direct product of rings and the trivial ring extensions.

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