Finitely generated projective modules over exchange rings
The authors study finitely generated projective modules over exchange rings. They prove that a finitely generated projective module over an exchange ring \(R\) is isomorphic to a direct sum of cyclic \(R\)-modules generated by idempotents. If \(R\) is an exchange ring such that \(R/J(R)\) is a ring with central idempotents, then the cancellation law holds in \(p(R)\), the category of all finitely generated projective \(R\)-modules. If \(R\) is an exchange ring with central idempotents then the family of all nonnegative integral-valued continuous functions on \(\text{Spec} (R)\) is precisely the family of all rank functions from \(\text{Spec} (R)\) into \(Z_ +\) induced by finitely generated projective \(R\)-modules. Finally, if \(R\) is an exchange ring with central idempotents, there is an order- preserving isomorphism between \((K_ 0 (R), [R])\) and \((H_ 0 (R), u)\), where \(K_ 0(R)\) is the Grothendieck group of \(R, H_ 0(R)\) is the set of all continuous functions from \(\text{Spec} (R)\) into \(Z\) and \(u\) is the order unit of \(H_ 0 (R)\).
scientific article; zbMATH DE number 733197