Rings with divisibility on ascending chains of ideals

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.