Let R be any ring The ascending chain condition (ACC) for ideals holds in R if every strictly increasing sequence N 1 N 2 N 3 of ideals in R is of finite length The maximum condition (MC) for ideals holds in R if every nonempty set S of ideals in R contains an ideal not properly contained in any ot...

We show that ACC implies MC that MC implies FBC and that FBC implies ACC ACC implies MC Suppose that ...

Let R be any ring. The ascending chain condition (ACC) for ideals holds in R if every

Question:

Let R be any ring. The ascending chain condition (ACC) for ideals holds in R if every strictly increasing sequence N₁ ⊂ N₂ ⊂ N₃ ⊂ · · · of ideals in R is of finite length. The maximum condition (MC) for ideals holds in R if every nonempty set S of ideals in R contains an ideal not properly contained in any other ideal of the set S. The finite basis condition (FBC) for ideals holds in R if for each ideal N in R, there is a finite set B_N = {b_i, · · · , b_n} ⊆ N such that N is the intersection of all ideals of R containing B_N. The set B_N is a finite generating set for N. Show that for every ring R, the conditions ACC, MC, and FBC are equivalent.