Projective modules over some Prüfer rings. (Q1419581)

A Prüfer ring \(R\) is considered here usually under the assumption that all its 0-divisors are contained in the Jacobson radical of \(R\) (smallness of 0-divisors). Frequently, the \(1\tfrac 12\) generator property [\textit{R. C. Heitmann} and \textit{L. S. Levy}, Rocky Mt. J. Math. 5, 361--373 (1975; Zbl 0371.13015)] is also assumed. Then it is established that the reduced Grothendieck group (of isomorphism classes of finitely generated projective modules over \(R)\) is isomorphic to the Picard group Pic\((R)\) and that any infinite direct sum of nonzero countably generated projective \(R\)-modules as well as any non countably generated projective \(R\)-module are free. In so doing, several known results concerning (not only Prüfer) domains are generalized to the case of arbitrary (commutative) rings.