Lemma 10.55.1. If $R$ is an Artinian local ring then the length function defines a natural abelian group homomorphism $\text{length}_ R : K'_0(R) \to \mathbf{Z}$.

Proof. The length of any finite $R$-module is finite, because it is the quotient of $R^ n$ which has finite length by Lemma 10.53.6. And the length function is additive, see Lemma 10.52.3. $\square$

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