-A separative ring is one whose finitely generated projective modules satisfy the property A EB A � A EB B � B EB 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 separate exchange rings is very broad, and, notably, closed under extensions of ideals by factor rings. That is, if an exchange ring R has an ideal I with I and Rf I both separative, then R is separative.
SEPARATIVE CANCELLATION FOR PROJECTIVE MODULES OVER EXCHANGE RINGSP. ARA, K.R. GOODEARL, K.C. O'MEARA AND E. PARDO ABSTRACT. A separative ring is one whose finitely generated projective modules satisfy the property A EB A ~ A EB B ~ B EB 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 ring R has an ideal I with I and R/ I both separative, then R is separative.
We show that every (discrete) group ring D½G of a free-by-amenable group G over a division ring D of arbitrary characteristic is stably finite, in the sense that one-sided inverses in all matrix rings over D½G are two-sided. Our methods use Sylvester rank functions and the translation ring of an amenable group. # 2002 Elsevier Science (USA)
For any (unital) exchange ring R whose finitely generated projective modules satisfy the separative cancellation property (, it is shown that all invertible square matrices over R can be diagonalized by elementary row and column operations. Consequently, the natural homomorphism GL 1 (R) → K 1 (R) is surjective. In combination with a result of Huaxin Lin, it follows that for any separative, unital C*-algebra A with real rank zero, the topological K 1 (A) is naturally isomorphic to the unitary group U(A) modulo the connected component of the identity. This verifies, in the separative case, a conjecture of Shuang Zhang.
I This paper characterizes products of idempotents in (von Neumann) regular rings which are unit-regular or right self-injective. For unit-regular rings, the minimum number of idempotents needed in such a product is determined, thereby generalizing a 1978 result of Ballantine in the case of a matrix with entries from a field.
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