On n-absorbing and strongly n-absorbing ideals of amalgamation

Issoual, Mohammed; Mahdou, Najib; Moutui, Moutu Abdou Salam

doi:10.1142/s0219498820501996

Cited by 7 publications

(2 citation statements)

References 13 publications

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“…The concept of n-absorbing ideals is a generalization of the concept of prime ideals (note that a prime ideal of R is a 1-absorbing ideal of R). For more details on n-absorbing ideals, we refer the reader to [11][12][13]. We investigate rings in which every n-absorbing ideal of R is a prime ideal, where n ≥ 2 is an integer, called n-AB rings.…”

Section: Introductionmentioning

confidence: 99%

On $n$-absorbing prime ideals of commutative rings

Issoual

Mahdou

Moutui

2022

Hacettepe Journal of Mathematics and Statistics

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This paper investigates the class of rings in which every n-absorbing ideal is a prime ideal, called n-AB ring, where n is a positive integer. We give a characterization of an n-AB ring. Next, for a ring R, we study the concept of Ω(R) = {ω R (I); I is a proper ideal of R}, where ω R (I) = min{n; I is an n-absorbing ideal of R}. We show that if R is an Artinian ring or a Prüfer domain, then Ω(R) ∩ N does not have any gaps (i.e., whenever n ∈ Ω(R) is a positive integer, then every positive integer below n is also in Ω(R)). Furthermore, we investigate rings which satisfy property (**) (i.e., rings R such that for each proper idealwhere M in R (I) denotes the set of prime ideals of R minimal over I). We present several properties of rings that satisfy condition (**). We prove that some open conjectures which concern n-absorbing ideals are partially true for rings which satisfy condition (**). We apply the obtained results to trivial ring extensions.

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Section: Introductionmentioning

confidence: 99%

On $n$-absorbing prime ideals of commutative rings

Issoual

Mahdou

Moutui

2022

Hacettepe Journal of Mathematics and Statistics

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“…Also, an ideal I of R is a semi-n-and to the Krull dimension. For more details on amalgamation rings, we refer the reader to [11], [14], [15].…”

Section: Introductionmentioning

confidence: 99%

ON $(m,n)$-CLOSED IDEALS IN AMALGAMATED ALGEBRA

Issoual

Mahdou

Moutui

2021

International Electronic Journal of Algebra

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Let R be a commutative ring with 1 = 0 and let m and n be integers with 1 ≤ n < m. A proper ideal I of R is called an (m, n)-closed ideal of R if whenever a m ∈ I for some a ∈ R implies a n ∈ I. Let f : A → B be a ring homomorphism and let J be an ideal of B. This paper investigates the concept of (m, n)-closed ideals in the amalgamation of A with B along J with respect f denoted by A f J. Namely, Section 2 investigates this notion to some extensions of ideals of A to A f J. Section 3 features the main result, which examines when each proper ideal of A f J is an (m, n)-closed ideal. This allows us to give necessary and sufficient conditions for the amalgamation to inherit the radical ideal property with applications on the transfer of von Neumann regular, π-regular and semisimple properties.

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Absorbing Ideals in Commutative Rings: A Survey

Badawi

2023

Algebraic, Number Theoretic, and Topological Aspects of Ring Theory

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On n-absorbing and strongly n-absorbing ideals of amalgamation

Cited by 7 publications

References 13 publications

On $n$-absorbing prime ideals of commutative rings

On $n$-absorbing prime ideals of commutative rings

ON $(m,n)$-CLOSED IDEALS IN AMALGAMATED ALGEBRA

Absorbing Ideals in Commutative Rings: A Survey

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