Questions: Valuation Rings

Question 1

Which of the following is a discrete valuation ring?

• A. The integers Z
• B. The localization Z_(p) for a prime p
• C. The polynomial ring k[x]
• D. The ring of integers Z[i] of the Gaussian integers

Answer: (B)

Z_(p) is a local PID with maximal ideal (p) -- it has exactly one nonzero prime ideal. Its fraction field is Q, and the p-adic valuation v_p measures divisibility by p. Z is not local (it has infinitely many maximal ideals). k[x] is not local. Z[i] is a Dedekind domain but not local.

Question 2

Every valuation ring is a local ring.

• T. True
• F. False

Answer: True

In a valuation ring V, the non-units form an ideal (if a and b are non-units, then say a/b is in V, so a = (a/b)b shows a is in (b); thus non-units are totally ordered, and their union is an ideal). The set of non-units is therefore the unique maximal ideal, making V local.

Question 3

Characterize discrete valuation rings among local domains (give three equivalent conditions).

Model answer: A local domain (R, m) is a DVR iff (1) R is a Noetherian valuation ring, iff (2) R is a regular local ring of dimension 1, iff (3) R is a local PID that is not a field.

These equivalences connect valuation theory to dimension theory and regularity. Condition (2) says m is generated by one element (the uniformizer π), and every nonzero ideal is (π^n) for some n. Condition (3) says the only ideals are (0) and (π^n). The value group is Z, with v(uπ^n) = n for units u.

Question 4

A valuation ring that is not a DVR is never Noetherian.

• T. True
• F. False

Answer: True

Among valuation rings, the Noetherian ones are exactly the DVRs (and fields, which are trivial valuation rings). If V is a valuation ring with value group not isomorphic to Z (or {0}), then V has ideals that are not finitely generated. For example, a valuation ring with value group Q has no finitely generated maximal ideal.

Question 5

Explain the role of DVRs in the theory of Dedekind domains.

Model answer: A Dedekind domain R can be characterized as a Noetherian domain whose localization R_P at every nonzero prime P is a DVR. The unique factorization of ideals in R corresponds to specifying the valuation v_P(I) at each prime P -- an ideal is determined by its 'order of vanishing' at each point.

This is the local-global principle for Dedekind domains. Globally, ideals factor as products of primes. Locally at each prime P, the localization R_P is a DVR with uniformizer π_P, and every ideal of R_P is (π_P^n). The exponent n = v_P(I) is the P-adic valuation of the ideal. The ideal I is recovered as the intersection of the local data: I = ∩_P {x ∈ K : v_P(x) ≥ v_P(I)}.