Rings in which every 2-absorbing ideal is prime

Bennis, Driss; Fahid, Brahim

doi:10.1007/s13366-017-0366-2

Cited by 9 publications

(4 citation statements)

References 8 publications

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“…Then P is idempotent. Moreover, if I is an ideal of R such that √ I = P, then I is an n-absorbing ideal of R for some positive integer if and only if I = P. (4) If P 2 ̸ = P , then there is no n-absorbing ideal of R between P and P 2 for every positive integer n. (5) The Conjecture three holds for every radical ideal of R. (6) The Conjecture one holds for every radical ideal of R. (7) Let n be a positive integer in Ω(R), then {1, • • • , n} ⊆ Ω(R).…”

Section: Rings Satisfying |Minmentioning

confidence: 99%

“…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. Note that the authors in [6] studied rings where every 2-absorbing ideal of R is prime.…”

Section: Introductionmentioning

confidence: 99%

“…Hence, Conjecture three holds for I. (6) Let I be a radical ideal of R which is n-absorbing. By assertion (5) above, we have…”

mentioning

confidence: 98%

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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: Rings Satisfying |Minmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On $n$-absorbing prime ideals of commutative rings

Issoual

Mahdou

Moutui

2022

Hacettepe Journal of Mathematics and Statistics

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“…Let us recall that a ring R is called 2-AB if every 2-absorbing ideal of R is prime [4,Definition 2.1]. Inspired by this, we give the following definition: Definition 3.1.…”

Section: -Ab Semiringsmentioning

confidence: 99%

On 2-absorbing ideals of commutative semirings

Behzadipour

Nasehpour

2019

J. Algebra Appl.

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In this paper, we investigate 2-absorbing ideals of commutative semirings and prove that if a is a nonzero proper ideal of a subtractive valuation semiring S then a is a 2-absorbing ideal of S if and only if a = p or a = p 2 where p = √ a is a prime ideal of S. We also show that each 2-absorbing ideal of a subtractive semiring S is prime if and only if the prime ideals of S are comparable and if p is a minimal prime over a 2-absorbing ideal a, then am = p, where m is the unique maximal ideal of S.2010 Mathematics Subject Classification. 16Y60; 13A15.

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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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Rings in which every 2-absorbing ideal is prime

Cited by 9 publications

References 8 publications

On $n$-absorbing prime ideals of commutative rings

On $n$-absorbing prime ideals of commutative rings

On 2-absorbing ideals of commutative semirings

Absorbing Ideals in Commutative Rings: A Survey

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