A discrete valuation ring is a local principal ideal domain that is not a field. Its unique nonzero prime ideal is generated by a uniformizer 
, and every nonzero element is
a unit times a nonnegative power of 
. The quotient by 
is its residue field.
Completeness refers to the topology defined by the powers 
. For a perfect field 
of characteristic 
, 
is a complete discrete valuation ring with uniformizer
.
Integral Donovan finiteness fixes such a coefficient ring before counting Morita types.
Discrete Valuation Ring
See also
Local Ring, Principal Ideal Domain, Witt VectorExplore with Wolfram|Alpha
References
Milne, J. S. "Algebraic Number Theory." Version 3.08, 19 Jul 2020. https://www.jmilne.org/math/CourseNotes/ANT.pdf.Cite this as:
Weisstein, Eric W. "Discrete Valuation Ring." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DiscreteValuationRing.html