Answer to Question #17370 in Algebra for Melvin Henriksen

Question

Show that the following are also equivalent:
(A) R is reduced (no nonzero nilpotents), and K-dim R = 0.
(B) R is von Neumann regular.
(C) The localizations of R at its maximal ideals are all fields.

Answer 1

We know that for acommutative 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.
Upon specializing to reduced rings,(1) becomes (A) and (2) becomes (B), so we have (A) ⇔ (B). The implication (A) ⇒ (C) is already done in (1) ⇒ (2) above, and (C)⇒(A) follows from a similar standard local-global argument in commutativealgebra.