On $n$-absorbing prime ideals of commutative rings

Abstract

This paper investigates the class of rings in which every $n$-absorbing ideal is a prime ideal, called $n$-AB ring, where $n$ is a positive integer. We give a characterization of an $n$-AB ring. Next, for a ring $R$, we study the concept of $\Omega \left(R\right)=\left\{{\omega }_R\right(I);I\text{is a proper ideal of}R\},$ where ${\omega }_R\left(I\right)=min\{n;I\text{is an}n\text{-absorbing}\text{ideal of}R\}$. We show that if $R$ is an Artinian ring or a Prüfer domain, then $\Omega \left(R\right)\cap N$ does not have any gaps (i.e., whenever $n\in \Omega \left(R\right)$ is a positive integer, then every positive integer below $n$ is also in $\Omega \left(R\right)$). Furthermore, we investigate rings which satisfy property (**) (i.e., rings $R$ such that for each proper ideal $I$ of $R$ with ${\omega }_R\left(I\right)<\infty$, $\omega_{R}(I)=\mid Min_R(I)\mid $, where $Mi{n}_R\left(I\right)$ denotes the set of prime ideals of $R$ minimal over $I$). We present several properties of rings that satisfy condition (**). We prove that some open conjectures which concern $n$-absorbing ideals are partially true for rings which satisfy condition (**). We apply the obtained results to trivial ring extensions.

Keywords

• n-absorbing ideal
• prime ideal
• primary ideal
• Noetherian ring
• Artinian ring
• Prüfer ring

References

1. [1] D.F. Anderson and A. Badawi, On n-absorbing ideals of commutative rings, Commun. Algebra, 39 (5), 1646-1672, 2011.
2. [2] D.D. Anderson and M. Winders, Idealization of a module, J. Commut. Algebra, 1 (1), 3-56, 2009.
3. [3] A. Badawi, On 2-absorbing ideals of commutative rings, Bull. Aust. Math. Soc. 75, 417-429, 2007.
4. [4] A. Badawi, M. Issoual and N. Mahdou, On n-absorbing ideals and (m, n)-closed ideals in trivial extension rings, J. Algebra Appl. 18 (7), 2019.
5. [5] C. Bakkari, S. Kabbaj and N. Mahdou, Trivial extension definided by Prûfer conditions, J. Pure App. Algebra 214, 53-60, 2010.
6. [6] D. Bennis and B. Fahid, Rings in which every 2-absorbing ideal is prime, Beitr. Algebra Geom. 59 (2), 391-396, 2018.
7. [7] H.S. Choi and A. Walker, The radical of an n-absorbing ideal, J. Commut. Algebra, 12 (2), 171-177, 2020.
8. [8] R.W. Gilmer, Ring in which every semi-primary ideals are primary, Pacific J. Math. 12 (4), 1273-1276, 1962.

9. [9] R.W. Gilmer and J. Leonard Mott, Multiplication rings as rings in which ideals with prime radical are primary, Trans. Amer. Math. Soc. 114, 40-52, 1965.
10. [10] S. Glaz, Commutative coherent rings, Springer-Verlag, Lecture Notes in Mathematics, 13-71, 1989.
11. [11] M. Issoual and N. Mahdou, Trivial extensions defined by 2-absorbing-like conditions, J. Algebra Appl. 17 (11), 1850208, 2018.
12. [12] M. Issoual, N. Mahdou and M.A.S. Moutui, On n-absorbing and strongly n-absorbing ideals of amalgamation, J. Algebra Appl. 19 (10) 2050199, 2020.
13. [13] M. Issoual, N. Mahdou and M.A.S. Moutui, On (m, n)-closed ideals in amalgamated algebra, Int. Electron. J. Algebra, 29, 134-147, 2021.
14. [14] S. Kabbaj, Matlis’ semi-regularity and semi-coherence in trivial ring extensions : a survey, Moroccan J. Algebra Geometry Appl. In Press.
15. [15] S. Kabbaj and N. Mahdou, Trivial extensions defined by coherent-like conditions, Comm. Algebra, 32 (10), 3937-3953, 2004.
16. [16] A. Laradji, On n-absorbing rings and ideals, Colloq. Math. 147 (2) 265-273, 2017.
17. [17] H.F. Moghimi and S.R. Naghani, On n-absorbing ideals and the n-Krull dimension of a commutative ring, J. Korean Math. Soc. 53 (6), 1225-1236, 2016.