On 2-absorbing ideals of commutative rings. (English) Zbl 1120.13004

Badawi, Ayman

Bull. Aust. Math. Soc. 75, No. 3, 417-429 (2007).

Summary: Suppose that \(R\) is a commutative ring with \(1\neq 0\). 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 \(a,b,c\in R\) and \(abc\in I\), then \(ab\in I\) or \(ac\in I\) or \(bc\in I\). It is shown that a nonzero proper ideal \(I\) of \(R\) is a 2-absorbing ideal if and only if whenever \(I_1 I_2 I_3\subseteq I\) for some ideals \(I_1\), \(I_2\), \(I_3\) of \(R\), then \(I_1I_2\subseteq I\) or \(I_2 I_3\subseteq I\) or \(I_1I_3\subseteq I\). It is shown that if \(I\) is a 2-absorbing ideal of \(R\), then either \(\text{Rad}(I)\) is a prime ideal of \(R\) or \(\text{Rad}(I)= P_1\cap P_2\) 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 \(M_1M_2\) 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 \(R\).

MSC:

13A15

Ideals and multiplicative ideal theory in commutative rings

References:

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[4]	Huckaba, Commutative rings with zero divisors (1988) · Zbl 0637.13001