Rings with divisibility on ascending chains of ideals

Authors : Oussama Aymane Es Safi, Najib Mahdou, Mohamed Yousif

Pages : 82-89

Doi:10.24330/ieja.1299720

View : 126 | Download : 174

Publication Date : 2024-01-09

Article Type : Research Paper

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

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