## 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.

### MSC:

 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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