The map of the Witt ring of a domain into the Witt ring of its field of fractions

Let R R be an integral domain with field of fractions K K . This paper studies the kernel of the map W ( R ) → W ( K ) W(R) → W(K) , where W W is the Witt ring functor. In case R R is regular and noetherian, it is shown that the kernel is a nilideal. The kernel is zero if R R is a complete regular local noetherian ring with 2 a unit. Examples are given to show that the regularity assumptions are needed.