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.


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


[1] DOI: 10.1080/00927870701724177 · Zbl 1140.13005
[2] DOI: 10.1142/S0219498810004221 · Zbl 1203.13001
[3] DOI: 10.1017/S0004972700039344 · Zbl 1120.13004
[4] Bastida , E. , Gilmer , R. ( 1973 ). Overrings and divisorial ideals in rings of the formD + M. Michigan Math. J.20 : 79 – 95 . · Zbl 0239.13001
[5] DOI: 10.1307/mmj/1029001619 · Zbl 0318.13007
[6] DOI: 10.1007/978-3-642-88405-4
[7] DOI: 10.1016/j.jpaa.2007.10.020 · Zbl 1133.20047
[8] Gilmer R., Multiplicative Ideal Theory (1992) · Zbl 0248.13001
[9] Huckaba J., Rings with Zero-Divisors (1988) · Zbl 0637.13001
[10] Larson M. D., Multiplicative Theory of Ideals (1971)
[11] Lucas , T. G. ( 2000 ). Examples built withD + M,A + XB[X] and other pullback constructions.Non-Noetherian Commutative Rings.Math. Appl., 520 . Dordrecht : Kluwer Acad. Publ. , pp. 341 – 368 . · Zbl 1005.13002
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