
Vietnam Journal of Mathematics 34:2(2006)
171178

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, WGAring) if for each idempotent e in R, there exist idempotents f,g,h
in R such that $eR\cong fR\oplus
gR$ and $(1e)R\cong fR\oplus hR$, while gR and hR have no
isomorphic nonzero summands. By an example we will show that the class of
generalized abelian rings (for short, GArings)
introduced in [10] is a proper subclass of the class of WGArings. 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 WGAring.


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

Keywords: K_{0}group,
exchange ring, weakly generalized Abelian ring, Stable range 1,
$\ell$group.


Established
by Vietnam Academy of Science and Technology & Vietnam Mathematical
Society
Published
by Springer since January 2013

