This is inspired by the theorem mentioned in 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.