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Vietnam Journal of Mathematics 35:2(2007) 153-162  $IBN_{2}$ Rings and Their Grothendieck Groups Xinmin Lu Abstract.  Following P.M. Cohn [1], a ring $R$ is called an $IBN_{2}$ ring if $R^{m}\cong R^{m}\oplus M$ then $M=0$. In this paper, we will establish some new characterizations for such class of rings. As an application, we further investigate the structure of their Grothendieck groups, which generalizes the main result Theorem 5.3 in [8].   2000 Mathematics Subject Classification: 16D90, 16E20, 06F20, 06F15. Keywords: $IBN_{2}$ ring; $K_{0}$-group; partially ordered abelian group; $\ell$-group.

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