On 2-absorbing ideals of commutative rings

Extract

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 I₁I₂I₃ ⊆ I for some ideals I₁,I₂,I₃ of R, then I₁I₂ ⊆ I or I₂I₃ ⊆ I or I₁I₃ ⊆ 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) = P₁ ⋂ P₂ where P₁,P₂ 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 M₂ for some maximal ideal M of R or M₁M₂ where M₁,M₂ 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 M₂ for some maximal ideal M of R or M₁M₂ where M₁,M₂ are some maximal ideals of R.