Rings with divisibility on ascending chains of ideals

Year 2024, Volume: 35 Issue: 35, 82 - 89, 09.01.2024

Oussama Aymane Es Safı Najib Mahdou Mohamed Yousıf

https://doi.org/10.24330/ieja.1299720

Abstract

According to Dastanpour and Ghorbani, a ring $R$ is said to satisfy divisibility on ascending chains of right ideals ($A C C_{d}$) if, for every ascending chain of right ideals $I_{1} \subseteq I_{2} \subseteq I_{3} \subseteq I_{4} \subseteq \ldots $ of $R$, there exists an integer $k \in \mathbb{N}$ such that for each $i \geq k$, there exists an element $a_{i} \in R$ such that $I_{i} =a_{i} I_{i +1}$. In this paper, we examine the transfer of the $A C C_{d}$-condition on ideals to trivial ring extensions. Moreover, we investigate the connection between the $A C C_{d}$ on ideals and other ascending chain conditions. For example we will prove that if $R$ is a ring with $A C C_{d}$ on ideals,\ then $R$ has $A C C$ on prime ideals.

Keywords

Commutative ring, ring with the $A c c_{d}$-condition, trivial ring extension, noetherian ring

References

• D. D. Anderson and M. Winders, Idealization of a module, J. Commut. Algebra, 1(1) (2009), 3-56.
• C. Bakkari, Rings in which every homomorphic image is a Noetherian domain, Gulf J. Math., 2 (2014), 1-6.
• C. Bakkari, S. Kabbaj and N. Mahdou, Trivial extensions defined by Prüfer conditions, J. Pure Appl. Algebra, 214(1) (2010), 53-60.
• C. Bakkari and N. Mahdou, On weakly coherent rings, Rocky Mountain J. Math., 44(3) (2014), 743-752.
• R. Dastanpour and A. Ghorbani, Rings with divisibility on chains of ideals, Comm. Algebra, 45(7) (2017), 2889-2898.
• S. Glaz, Commutative Coherent Rings, Lecture Notes in Mathematics, 1371, Springer-Verlag, Berlin, 1989.
• J. A. Huckaba, Commutative Rings with Zero Divisors, Monographs and Textbooks in Pure and Applied Mathematics, 117, Marcel Dekker, Inc., New York, 1988.
• S. Kabbaj and N. Mahdou, Trivial extensions defined by coherent-like conditions, Comm. Algebra, 32(10) (2004), 3937-3953.
• B. L. Osofsky, A generalization of quasi-Frobenius rings, J. Algebra, 4 (1966), 373-387.