ON EULER CHARACTERISTIC OF MODULES~(**)

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.

ON EULER CHARACTERISTIC OF MODULES

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 $\\chi (M \otimes N) = \chi (M)\chi (N)\]$, where $\\chi \]$ denotes the Euler characteristic. (2) If $\f:{K_0}(R) \to Z\]$ is a ring isomorphism, where $\{K_0}(R)\]$ denotes the Grothendieck group of R, $\{K_0}(R)\]$ is a ring when R is comnmtative, then $\f(M]) = \chi (M)\]$ and $\\chi (M \otimes N) = \chi (M)\chi (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.