On 2-absorbing ideals of commutative rings

By Ayman Badawi. Bulletin of the Australian Mathematical Society, 2007

302 citations recorded by the source corpus.

Suppose that R is a commutative ring with 1 ≠ 0. In this paper, we introduce the concept of 2-absorbing ideal which is a generalisation of prime ideal. A nonzero proper ideal I of R is called a 2-absorbing ideal of R if whenever a, b, c ∈ R and abc ∈ I, then ab ∈ I or ac ∈ I or bc ∈ I. It is shown that a nonzero proper ideal I of R is a 2-absorbing ideal if and only if whenever I1I2I3 ⊆ I for some ideals I1,I2,I3 of R, then I1I2 ⊆ I or I2I3 ⊆ I or I1I3 ⊆ I. It is shown that if I is a 2-absorbing ideal of R, then either Rad(I) is a prime ideal of R or Rad(I) = P1 ⋂ P2 where P1,P2 are the only distinct prime ideals of R that are minimal over I. Rings with the property that every nonzero proper ideal is a 2-absorbing ideal are characterised. All 2-absorbing ideals of valuation domains and Prüfer domains are completely described. It is shown that a Noetherian domain R is a Dedekind domain if and only if a 2-absorbing ideal of R is either a maximal ideal of R or M2 for some maximal ideal M of R or M1M2 where M1,M2 are some maximal ideals of R. If RM is Noetherian for each maximal ideal M of R, then it is shown that an integral domain R is an almost Dedekind domain if and only if a 2-absorbing ideal of R is either a maximal ideal of R or M2 for some maximal ideal M of R or M1M2 where M1,M2 are some maximal ideals of R.

Subfields: Algebra and Number Theory, Rehabilitation

View Paper: https://www.semanticscholar.org/paper/5759512a8cf9efd876cee9bcdaccf6cef52ac373

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Citations in the corpus

2-Absorbing and Weakly 2-Absorbing Submodules (2012, background): “Badawi in [6] generalized the concept of prime ideals in a different way.”
Generalizations of Prime Ideals (2008, background): “Badawi (2007) defined a proper ideal I of R to be absorbing if a1a2a3 ∈ I a1 a2 a3 ∈ R implies a1a2 ∈ I , or a1a3 ∈ I , or a2a3 ∈ I .”
S-prime ideals of a commutative ring (2019, background): “There are several ways to generalize the notion of a prime ideal [see, Galovich (1978); Bhatwadekar and Sharma (2005); Badawi (2007)].”
On n-Absorbing Ideals of Commutative Rings (2011, background): “These results generalize corresponding results for 2-absorbing ideals in [3].”
ON 2-ABSORBING PRIMARY IDEALS IN COMMUTATIVE RINGS (2014, background): “The concept of 2-absorbing ideal, which is a generalization of prime ideal, was introduced by Badawi in [3] and studied in [2], [8], and [4].”