Question #17370

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.

Expert's answer

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.

(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)

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