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Vietnam Journal of Mathematics 34:2(2006) 171-178

$K_{0}$ of Exchange Rings with Stable Range 1

Xinmin Lu and Hourong Qin

Abstract. A ring $R$ is called weakly generalized abelian (for short, $WGA$-ring) if for each idempotent $e$ in $R$, there exist idempotents $f,g,h$ in $R$ such that $eR\cong fR\oplus gR$ and $(1-e)R\cong fR\oplus hR$, while $gR$ and $hR$ have no isomorphic nonzero summands. By an example we will show that the class of generalized abelian rings (for short, $GA$-rings) introduced in [10] is a proper subclass of the class of $WGA$-rings. We will prove that, for an exchange ring $R$ with stable range 1, $K_{0}(R)$ is an $\ell$-group if and only if $R$ is a $WGA$-ring.

2000 Mathematics Subject Classification: 19A49, 16E20, 06F15.

Keywords: $K_0$-group, exchange ring, weakly generalized Abelian ring, Stable range 1, $\ell$-group.