Valuation Rings, Dominant Local Rings

Dominant Local Rings

Let R and S be local rings in the field F. We say S dominates R if S contains R, and the maximal ideal of S contains the maximal ideal of R. This latter condition is equivalent to saying the maximal ideal of S intersected with R gives the maximal ideal of R. In other words, the nonunits of S, intersect R, produce the nonunits of R - just as the units of S, intersect R, produce the units of R.

Verify that dominance is a partial ordering.

Let C be an ascending chain of dominant local rings, and let U be their union. Two nonunits in U are nonunits in some ring R in the chain. Remember that R is a local ring. Let z be their sum. If z is a unit in U then 1/z appears in some ring S in the chain. Now z and 1/z are units in S. Intersect S and R, and if z is common to both, and a nonunit in R, it should be a nonunit in S. This is a contradiction, hence z is a nonunit in U. The sum of nonunits remains a nonunit.

Similarly we can multiply a nonunit in U by anything else in U and obtain a nonunit, so the nonunits form an ideal, and U is local.

Intersect U with any ring R in the chain, and the nonunits remain nonunits. Therefore U dominates the entire chain. We have found a local ring that dominates C. Every ascending chain of local rings is bounded by a local ring, i.e. the union, that dominates the entire chain. By zorn's lemma there is always a maximal local ring in F, that dominates any given local ring R, though this maximal ring may be F itself.

Such is the case when F is the union of all finite fields of characteristic p. If R is a ring inside F, and x is an element of R, then Zp adjoin x creates a finite field in R, every x is invertible, and R is a field. All the rings in F are fields.

Every subring of F is either F itself, or a finite field. Fields are ordered by containment, and each is technically a local ring, with 0 acting as the maximal ideal.

Now start with any finite field K in F and build a chain of ascending finite fields, a chain of local rings, each dominating the previous. The "maximal" local ring is the entire field F.

We are going to prove that a maximal local ring (with respect to dominance) is a valuation ring, but first we need a lemma.