Build the idea in three steps
- A discrete valuation ring is an integral domain with three certificates at once: every ideal is one generator’s multiples (principal ideal ring), the world is local (one gate), and yet it is not a field — the gate is real: maximalIdeal R ≠ ⊥.
- Choose a uniformizer π generating the maximal ideal. Every nonzero element is a unit times π^n for one natural number n ≥ 0. Zero has no such finite expression; valuation language treats its value as infinite. The earlier λ-count is an example of this kind of divisibility measurement.
- The summit world 𝒪 in the R = T statement is a DVR: [IsDomain 𝒪] and [IsDiscreteValuationRing 𝒪] appear side by side in the hypothesis row. Local plus principal means a single measurable defect — the language in which deformation rings are compared and patched together.
A small example
The Eisenstein integer ring itself is not local. Localizing it at the prime ideal generated by λ gives a DVR, where λ supplies the ruler. Counting how often λ divides a nonzero c illustrates the valuation used to rank the cube-case candidates.