## 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

### Keywords:

2-absorbing ideal; $$n$$-absorbing ideal; prime; Prüfer domain
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### References:

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