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On \(n\)-absorbing ideals of commutative rings. (English) Zbl 1232.13001
Let \(R\) be a commutative ring with \(1 \neq 0\) and \(n\) a positive integer. In this article, the authors study two generalizations of prime ideals: \(n\)-absorbing ideals and strongly \(n\)-absorbing ideals. A proper ideal \(I\) of \(R\) is called an \(n\)-absorbing ideal if whenever \(x_1\cdots x_{n+1} \in I\) for \(x_1 , \dots, x_{n+1} \in R\), then there are \(n\) of the \(x_i\)’s whose product is already in \(I\). A proper ideal \(I\) of \(R\) is called a strongly \(n\)-absorbing ideal if \(I_1 \cdots I_{n+1} \subset I\) for \(I_1 , \dots, I_{n+1} \subset R\), then there are \(n\) of the \(I_j\)’s whose product is already in \(I\).
The authors give many basic properties of \(n\)-absorbing ideals. It is shown that an \(n\)-absorbing ideal has at most \(n\) minimal prime ideals. The authors study the behalf of \(n\)-absorbing ideals with respect to various ring-theoretic constructions such as localization, factor rings, and idealization. Furthermore, \(n\)-absorbing ideals in several classes of commutative rings are studied and some good results are obtained. For example, in a noetherian ring every proper ideal is an \(n\)-absorbing ideal for some positive integer \(n\), and in a Prüfer domain, an ideal is an \(n\)-absorbing ideal for some integer \(n\) if and only if it is a product of prime ideals. At the final, the authors shown that, over Prüfer domains, the notion of \(n\)-absorbing ideals and the notion strongly \(n\)-absorbing ideals are equivalent.

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