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

Badawi, Ayman; Tekir, Unsal; Yetkin, Ece

Bull. Korean Math. Soc. 51, No. 4, 1163-1173 (2014).

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.

Reviewer: Ahmad Yousefian Darani (Ardebil)

MSC:

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

Keywords:

primary ideal; prime ideal; 2-absorbing ideal; \(n\)-absorbing ideal

Citations:

Zbl 1120.13004