There is a similar construction with "'Z " "'p " replaced by any complete discrete valuation ring with finite uniformizer.
12.
Each such valuation ring is a discrete valuation ring and its maximal ideal is called a " place " of " K / k ".
13.
There is a similar construction with "'Z " "'p " replaced by any complete discrete valuation ring with finite residue class field.
14.
Just as a Dedekind domain is locally a discrete valuation ring, a Pr�fer domain is locally a valuation ring, so that Pr�fer domains act as non-noetherian analogues of Dedekind domains.
15.
Let the discrete valuation ring " R " be the ring of formal power series over " K " whose coefficients generate a finite extension of " k ".
16.
Definitely a field is a valuation ring and not a DVR, and is * way * simpler than any non-discrete valuation ring ( valuation ring with non-discrete value group ).
17.
A Cohen ring is a field or a complete characteristic zero discrete valuation ring whose maximal ideal is generated by a prime number " p " ( equal to the characteristic of the residue field ).
18.
A value group is called " discrete " if it is isomorphic to the additive group of the integers, and a valuation ring has a discrete valuation group if and only if it is a discrete valuation ring.
19.
Geometry of degeneration is largely then about the passage from generic to special fibers, or in other words how specialization of parameters affects matters . ( For a discrete valuation ring the topological space in question is the Sierpinski space of topologists.
20.
Arguing as in Example 3 . above and using the Zariski-Samuel theorem, it is easy to check that Hungerford's theorem is equivalent to the statement that any special principal ring is the quotient of a discrete valuation ring.
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