Let R be an integral domain with field of fractions K. This paper studies the kernel of the map W(R) -W(K), where W is the Witt ring functor. In case R is regular and noetherian, it is shown that the kernel is a nilideal. The kernel is zero if R is a complete regular local noetherian ring with 2 a unit. Examples are given to show that the regularity assumptions are needed. If R is a domain we write K for K(0). We shall also write R for the set of squares in R. By a space E over R, we mean a

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... tely generated projective R-module together with a symmetric nondegenerate bilinear form <£. If F is a space over R then [E] denotes its equivalence class in W(R). We shall write (flj, • • • , a ) to denote the space E which is a free R-module with basis