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 (ACCd) if, for every ascending chain of right ideals I₁ ⊆ I₂ ⊆ I₃ ⊆ I₄ ⊆ ... of R, there exists an integer k ∈ N such that for each i ≥ k, there exists an element a_i ∈ R such that I_i = a_iI_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. [ABSTRACT FROM AUTHOR]