Question

For a commutative ring R, show that the following are equivalent:
(1) R has Krull dimension 0.
(2) rad R is nil and R/rad R is von Neumann regular.
(3) For any a ∈ R, the descending chain Ra ⊇ Ra2 ⊇ . . . stabilizes.
(4) For any a ∈ R, there exists n ≥ 1 such that an is regular.

Answer 1

(1) ⇒ (2). By (1), rad R is the intersection of all prime ideals, so itis nil. For the rest, we may assume that rad R = 0. For any a ∈ R, it suffices to show that Ra = Ra².Let p be any prime ideal of R. Since R is reduced, so is Rp.But pRp is the only prime ideal of Rp, so we have pRp = 0.Therefore, Rp is a field. In particular, Rpa = Rpa².It follows that (Ra/Ra²)p = 0 for every prime ideal p ⊂ R; hence Ra = Ra², asdesired.
(2) ⇒ (3). Let a ∈ R. By (2 ),a = a²b ∈ R/rad R for some b ∈ R, so (a − a²b)n =0 for some n ≥ 1. Expanding the LHS and transposing, we get aⁿ∈ Raⁿ⁺¹, and hence Raⁿ = Raⁿ⁺¹= · · · .
(3) ⇒ (4) is clear.
(4) ⇒ (1). Let p be any prime ideal, and a is not in p. By (4), aⁿ= a²ⁿb for some b ∈ R and some n ≥ 1. Then aⁿ(1− aⁿb) = 0 implies that 1 − aⁿb ∈ p. This shows that R/p is a field, so p is a maximal ideal.