For a separable odd degree field extension K/F the kernel of a Scharlau transfer of Witt rings s* : WK -» WF is a JΓF-module. We compute the prime ideals attached to ker s* and deduce that WK is not a projective WF -module if an ordering on F extends uniquely to K. An example shows WK may be a free WF-moάvXt if F is real and no ordering extends uniquely. For non-real, non-rigid F we show that K/F Galois and WK noetherian implies WK is not a projective WF-moάvle. If K/F is a finite extension of fields (characteristic not 2) then each non-trivial linear functional s: K —> F induces a Scharlau transfer s*: WK -> PFF on the Witt rings. When K = F(y/d) the kernel and image of s* are well known. We restrict our attention to separable odd degree extensions, where s* is surjective but little is known of ker s*. The map induced by inclusion r* : WF — > WK is injective and we view WF as a subring of WK. Then WK and ker s* are PFF-modules and our approach is module theoretic. WF need not be noetherian and ker s* need not be finitely generated over WF. So the usual theory of prime ideals associated to modules must be replaced by the notion of attached primes (in the sense of Dutton). We show no P(a, p) is attached to ker s*, P(a) is attached iff a has more than one extension to K and IF is attached iff WtKπ ker s* Φ 0. As a consequence, WK — WF iff each ordering on F extends uniquely to K and WtKr\kQrs* = 0. Another consequence is that WK is finitely generated over WF if F has only finitely many orderings and IF is not attached to ker s*. The main result deduced from the work on attached primes is that WK is not a projective WF -module if some ordering on F extends uniquely to K. WK may be projective, however, if F is real and no ordering extends uniquely. We present an example where K/F is Galois, F is real, both WK and WF noetherian rings and WK is a free WF-module. When F is non-real and non-rigid we show the same conditions (K/F Galois, WK and WF noetherian) implies WK is not a free WF-module. Weaker results hold under fewer restrictions on K/F.