This is inspired by the theorem mentioned in [Why is this theorem attributed to Serre?](https://mathoverflow.net/questions/99916/why-is-this-theorem-attributed-to-serre). But I'm not sure if it's research level. If not, please feel free to vote for closing. 


Let $R$ be a ring and let $Mod\_R$ be the category of finitely generated $R$-modules. 


What are examples of additive integer-valued functions on $Mod\_R$, i.e. functions $\lambda: Mod\_R \to \mathbb{Z}$ satisfying $\lambda(M) = \lambda(M') + \lambda(M'')$ for short exact sequences $$0 \to M' \to M \to M'' \to 0$$ in $Mod\_R$ ? 


Two obvious examples that come into my mind are: 


1. $\lambda(M)=\dim\_k M$ if $R=k$ is a field.
2. $\lambda(M)=\text{length}(M)$ if $R$ is Artinian.



Additive integer-valued functions on the module category 
Such functions are the same as homomorphisms $G\_0(R)\rightarrow\mathbb{Z}$ from the Grothendieck group of your category, the $G$-theory group of degree $0$. The answer is only trivial from this formal point of view. The computation of $G\_0(R)$ is non-trivial in general. If your ring is commutative noetherian and regular then $G\_0(R)=K\_0(R)$ is the $K$-theory group of degree $0$, i.e. additive functions only depend on the behaviour on projectives.


Let me complete my answer with the examples you consider in your question. If $R=k$ is a field $G\_0(k)=K\_0(k)=\mathbb{Z}$ generated by the isomorphism class of $k$, therefore all additive functions are multiples of the dimension. If $R$ is artinian then $G\_0(R)$ is the free abelian group on simple $R$-modules, hence not all additive functions are multiples of the length in general, but for each simple module $S$, $\lambda(S)=n\_S\cdot\operatorname{length}(S)$, and any choice of such $n\_S$ determines an additive function $\lambda$.


