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

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
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### References:

 [1] Larson, Multiplicative theory of ideals (1971) [2] Badawi, Houston J. Math. 26 pp 473– (2000) [3] Gilmer, Multiplicative ideal theory 90 (1992) [4] Huckaba, Commutative rings with zero divisors (1988) · Zbl 0637.13001
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