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\(n\)-absorbing ideals of commutative rings and recent progress on three conjectures: a survey. (English) Zbl 1390.13005

Fontana, Marco (ed.) et al., Rings, polynomials, and modules. Proceedings of the conferences “Recent advances in commutative ring and module theory”, Bressanone/Brixen, Italy, June 13–17, 2016 and “Conference on rings and polynomials”, Graz, Austria, July 3–8, 2016. Cham: Springer (ISBN 978-3-319-65872-8/hbk; 978-3-319-65874-2/ebook). 33-52 (2017).
Summary: Let \(R\) be a commutative ring with \(1\neq 0\). Recall that a proper ideal \(I\) of \(R\) is called a 2-absorbing ideal of \(R\) if \(a,b,c \in R\) and \(abc \in I\), then \(ab \in I\) or \(ac \in I\) or \(bc \in I\). A more general concept than 2-absorbing ideals is the concept of \(n\)-absorbing ideals. Let \(n \geq 1\) be a positive integer. A proper ideal \(I\) of \(R\) is called an \(n\)-absorbing ideal of \(R\) if \(a_1,a_2,\dots,a_{n+1} \in R\) and \(a_1a_2\cdots a_{n+1} \in I\), then there are \(n\) of the \(a_i\)’s whose product is in \(I\). The concept of \(n\)-absorbing ideals is a generalization of the concept of prime ideals (note that a prime ideal of \(R\) is a 1-absorbing ideal of \(R\)). In this survey article, we collect some old and recent results on \(n\)-absorbing ideals of commutative rings.
For the entire collection see [Zbl 1387.13003].

MSC:

13A15 Ideals and multiplicative ideal theory in commutative rings
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