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Vietnam Journal of Mathematics 34:2(2006)
171-178
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K0
of Exchange Rings with Stable Range 1
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Xinmin Lu and
Hourong Qin
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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, K0(R) is an $\ell$-group if and only if
R is a WGA-ring.
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2000 Mathematics Subject Classification: 19A49, 16E20,
06F15.
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Keywords: K0-group,
exchange ring, weakly generalized Abelian ring, Stable range 1,
$\ell$-group.
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Established
by Vietnam Academy of Science and Technology & Vietnam Mathematical
Society
Published
by Springer since January 2013
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