For several classes of commutative rings R it is known that every finitely generated projective R-modules is isomorphic to a direct sum of a free w x R-module and an invertible ideal of R. For instance, Steinitz 27 essenw x tially proved this for Dedekind domains. Serre 25 proved the same result for noetherian R of Krull dimension at most one. Our main result is that the same decomposition of projective modules holds over Witt rings. We Ε½ . note that Witt rings are not domains with two exceptions and are often far from noetherian.
Ε½ . Two sample consequences are: a every finitely generated projective Ε½ . module over a Witt ring R is free iff the reduced stability index st R F 2; Ε½ . b if I is a finitely generated ideal of a Witt ring and I contains an odd dimensional form then I can be generated by two elements. We also examine the case where R s WF is the Witt ring of a field F, F ; K is an odd degree extension, and so WK is a WF-module. We show that if F has only finitely many orderings and WK is finitely generated WF-projective then WK is in fact WF-free.
Ε½ From this point on, R denotes a Witt ring to be precise, an abstract w x. Witt ring as defined by Marshall 20 . If R has no orderings then R is local and every finitely generated projective module is free. So we assume w x Ε½ throughout that R is real. Serre's result 25 , mentioned above Swan's w x . formulation 28, p. 176 is the most helpful , implies that if R has only finitely many orderings then again our main result holds: finitely generated projectives are the direct sum of a free module and an invertible ideal. Thus the emphasis here is on the case where R has infinitely many orderings. The basic tool is a decomposition theorem for invertible ideals Ε½ w x. which is the usual primary decomposition cf. 5 if R has only finitely many orderings.