n-Absorbing Ideals of Commutative Rings and Recent Progress on Three Conjectures: A Survey

Ayman Badawi ()
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Ayman Badawi: American University of Sharjah, Department of Mathematics and Statistics

A chapter in Rings, Polynomials, and Modules, 2017, pp 33-52 from Springer

Abstract: Abstract Let 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 a 1, a 2, …, a n+1 ∈ R and a 1 a 2⋯a n+1 ∈ I, then there are n of the a i ’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.

Keywords: Prime; Primary; Weakly prime; Weakly primary; 2-Absorbing; n-Absorbing; Weakly 2-absorbing; Weakly n-absorbing; 2-Absorbing primary; Weakly 2-absorbing primary; 13A15 (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/978-3-319-65874-2_3

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