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

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 a1, a2, …, a n+1 ∈ R and a1a2⋯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.