n-Absorbing Ideals of Commutative Rings and Recent Progress on Three Conjectures: A Survey
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DescriptionLet R be a commutative ring with 1 ≠ 0. Recall that a proper ideal I of R is called a 2-absorbing ideal of R if a,b,c ∈ R and abc ∈ I, then ab ∈ I or ac ∈ I or bc ∈ I . A more general concept than 2-absorbing ideals is the concept of n-absorbing ideals. Let n ≥ 1 be a positive integer. A proper ideal I of R is called an n-absorbing ideal of R if a1,a2,…an+1 ∈ R and a1a2…an+1 ∈ I, then there are n of the ai's whose product is in I. 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). In this survey article, we collect some old and recent results on n-absorbing ideals of commutative rings.
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Badawi, Ayman; Celikel, Ece Yetkin (World Scientific, 2019-04-30)Let R be a commutative ring with nonzero identity. In this paper, we introduce the concept of 1-absorbing primary ideals in commutative rings. A proper ideal I of R is called a 1-absorbing primary ideal of R if whenever ...
Badawi, Ayman; Celikel, Ece Yetkin (World Scientific, 2022)Let R be a commutative ring with 1 ≠ 0. In this paper, we introduce the concept of weakly 1-absorbing primary ideal which is a generalization of 1-absorbing primary ideal. A proper ideal I of R is called a weakly 1-absorbing ...