In this case, it is either a field or it has exactly one non-zero prime ideal; in the latter case it is called a discrete valuation ring . ( By convention, a field is not a discrete valuation ring .)
22.
In this case, it is either a field or it has exactly one non-zero prime ideal; in the latter case it is called a discrete valuation ring . ( By convention, a field is not a discrete valuation ring .)
23.
Here is an example of a discrete valuation ring " A " of dimension 1 and characteristic " p " > 0 which is J-2 but not a G-ring and so is not quasi-excellent.
24.
It is a discrete valuation ring; the " unique " irreducible element is " X " and the valuation assigns to each function " f " the order ( possibly 0 ) of the zero of " f " at 0.
25.
For each closed point " x " of " X " we can consider the local ring " R " " x " at this point, which is a discrete valuation ring whose spectrum has one closed point and one open ( generic ) point.
26.
Lichtenbaum was an undergraduate at Harvard University ( bachelor's degree " summa cum laude " in 1960 ), where he also obtained his Ph . D . in 1964 ( Curves over discrete valuation rings, American Journal of Mathematics Bd . 90, 1968, S . 380-405 ).
27.
I am saying that " not-DVR " and " non-discrete valuation ring " are different, that is, " not " is not associative . " VR but Not ( DVR ) " is different than " ( not-D ) VR ", but only in a silly technical sense.
28.
In both cases, the hardest part of Cohen's proof is to show that the complete Noetherian local ring contains a "'coefficient ring "'( or "'coefficient field "'), meaning a complete discrete valuation ring ( or field ) with the same residue field as the local ring.
29.
This has an algebraic resemblance with the concept of a uniformizing parameter ( or just "'uniformizer "') found in the context of discrete valuation rings in commutative algebra; a uniformizing parameter for the DVR ( " R, m " ) is just a generator of the maximal ideal " m ".
30.
Conversely, the valuation \ nu : A \ rightarrow \ Z \ cup \ { \ infty \ } on a discrete valuation ring A can be extended in a unique way to a discrete valuation on the quotient field K = \ text { Quot } ( A ); the associated discrete valuation ring \ mathcal { O } _ K is just A.
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