On 2-absorbing ideals of commutative rings

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 I 1 I 2 I 3 ⊆ I for some ideals I 1 , I 2 , I 3 of R , then I 1 I 2 ⊆ I or I 2 I 3 ⊆ I or I 1 I 3 ⊆ 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 ) = P 1 ⋂ P 2 where P 1 , P 2 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 M 2 for some maximal ideal M of R or M 1 M 2 where M 1 , M 2 are some maximal ideals of R . If R M 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 M 2 for some maximal ideal M of R or M 1 M 2 where M 1 , M 2 are some maximal ideals of R .