RINGS WITH DIVISIBILITY ON ASCENDING CHAINS OF IDEALS

According to Dastanpour and Ghorbani, a ring R is said to satisfy divisibility on ascending chains of right ideals (ACC(d)) if, for every ascending chain of right ideals I-1 subset of I-2 subset of I-3 subset of I-4 subset of ... of R, there exists an integer k is an element of N such that for each i > k, there exists an element ai E R such that I-i = a(i)I(i+1). In this paper, we examine the transfer of the ACC(d)-condition on ideals to trivial ring extensions. Moreover, we investigate the connection between the ACC(d) on ideals and other ascending chain conditions. For example we will prove that if R is a ring with ACC(d) on ideals, then R has ACC on prime ideals.