摘要
This paper gives a characteristic property of the Euler characteristic for IBN rings. The following results: are proved. （1） If R is a commutative ring, M, N are two stable free R-modules, then χ（MN）=χ（M）χ（N）, where χ denotes the Euler characteristic. （2） If f: K<sub>0</sub>（R）→Z is a ring isomorphism, where K<sub>0</sub>（R） denotes the Grothendieck group of R, K<sub>0</sub>（R） is a ring when R is commutative, then f（[M]）=χ（M） and χ（MN）=χ（M）χ（N） when M, N are finitely generated projective R-modules, where.the isomorphism class [M] is a generator of K<sub>0</sub>（R）. In addition, some applications of the results above are also obtained.
This paper gives a characteristic property of the Euler characteristic for IBN rings. The following results: are proved. (1) If R is a commutative ring, M, N are two stable free R-modules, then χ(MN)=χ(M)χ(N), where χ denotes the Euler characteristic. (2) If f: K_0(R)→Z is a ring isomorphism, where K_0(R) denotes the Grothendieck group of R, K_0(R) is a ring when R is commutative, then f([M])=χ(M) and χ(MN)=χ(M)χ(N) when M, N are finitely generated projective R-modules, where.the isomorphism class [M] is a generator of K_0(R). In addition, some applications of the results above are also obtained.
基金
The Project supported by National Natural Science Foundation of China.
