On n-Absorbing Ideals of Commutative Rings

By David F. Anderson, Ayman Badawi. 2011

197 citations recorded by the source corpus.

Let R be a commutative ring with 1 ≠ 0 and n a positive integer. In this article, we study two generalizations of a prime ideal. A proper ideal I of R is called an n-absorbing (resp., strongly n-absorbing) ideal if whenever x 1…x n+1 ∈ I for x 1,…, x n+1 ∈ R (resp., I 1…I n+1 ⊆ I for ideals I 1,…, I n+1 of R), then there are n of the x i 's (resp., n of the I i 's) whose product is in I. We investigate n-absorbing and strongly n-absorbing ideals, and we conjecture that these two concepts are equivalent. In particular, we study the stability of n-absorbing ideals with respect to various ring-theoretic constructions and study n-absorbing ideals in several classes of commutative rings. For example, in a Noetherian ring every proper ideal is an n-absorbing ideal for some positive integer n, and in a Prüfer domain, an ideal is an n-absorbing ideal for some positive integer n if and only if it is a product of prime ideals.

Subfields: Algebra and Number Theory, Rehabilitation

View Paper: https://www.semanticscholar.org/paper/b121b48e02579c14a0c86fcec51535b40274bdf1

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Citations in the corpus

2-Absorbing and Weakly 2-Absorbing Submodules (2012, background): “Anderson and Badawi [8] generalized the concept of 2-absorbing ideals to n-absorbing ideals.”
On weakly 2-absorbing ideal of commutative rings (2016, background): “ncept of prime ideals in a diﬀerent way. He deﬁned a nonzero proper ideal I of R to be a 2-absorbing ideal of R if whenever a,b,c ∈ R and abc ∈ I, then ab ∈ I or ac ∈ I or bc ∈ I. Anderson and Badawi [3] generalized the concept of 2-absorbing ideals to n-absorbing ideals. According to their deﬁnitio”
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].”