Separative cancellation for projective modules over exchange rings

A separative ring is one whose finitely generated projective modules satisfy the propertyA⊕A⋟A⊕B⋟B⊕B⇒A⋟B. This condition is shown to provide a key to a number of outstanding cancellation problems for finitely generated projective modules over exchange rings. It is shown that the class of separative exchange rings is very broad, and, notably, closed under extensions of ideals by factor rings. That is, if an exchange ringR has an idealI withI andR/I both separative, thenR is separative. The research of the first and fourth authors was partially supported by a grant from the DGICYT (Spain) and by the Comissionat per Universitats i Recerca de la Generalitat de Catalunya. That of the second author was partially supported by a grant from the NSF (USA). The final version of this paper was prepared while he was visiting the Centre de Recerca Matemàtica, Institut d'Estudis Catalans in Barcelona, and he thanks the CRM for its hospitality.