On 2-absorbing primary ideals in commutative rings. (English) Zbl 1308.13001

A. Badawi [Bull. Aust. Math. Soc. 75, No. 3, 417–429 (2007; Zbl 1120.13004)] generalized the concept of prime ideals in commutative rings. According to his definition, a nonzero proper ideal \(I\) of a commutative ring \(R\) is said to be 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 \(bc\in I\). The paper under review is devoted to give a generalization of \(2\)-absorbing ideals. A proper ideal \(I\) of \(R\) is called a \(2\)-absorbing primary ideal of \(R\) if whenever \(a, b, c\in R\) and \(abc\in I\), then \(ab\in I\) or \(ac\in \sqrt{I}\) or \(bc\in \sqrt{I}\). In the paper under review, the authors among the other results prove that if I is a \(2\)-absorbing primary ideal of \(R\), then \(\sqrt{I}\) is a \(2\)-absorbing ideal of \(R\); they also prove that the product and intersection of \(2\)-absorbing primary ideals is again \(2\)-absorbing primary. They give some characterizations of \(2\)-absorbing primary ideals in Dedekind domains. They also give a number of suitable examples to clarify this class of ideals.


13A15 Ideals and multiplicative ideal theory in commutative rings
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
13G05 Integral domains


Zbl 1120.13004
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