Projective modules over Witt rings (Q1921912)

Let \(R\) denote an abstract Witt ring (in the sense of Marshall). The main result of the paper states that every finitely generated projective \(R\)-module is isomorphic to a direct sum of a free \(R\)-module and an invertible ideal of \(R\). (A similar result is known for Dedekind domains and for certain commutative Noetherian rings). As a consequence of this fact the author shows that every finitely generated projective module over \(R\) is free iff the reduced stability index of \(R\) does not exceed two. Moreover some other consequences are also derived. For instance: Every finitely generated ideal of \(R\) containing an odd dimensional form can be generated by two elements. The usual \(K_0\) group of \(R\) is isomorphic to a direct product of the group of integers and the Picard group of \(R\).