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 $Mod_R$ ?
1. $\lambda(M)=\dim_k M$ if $R=k$ is a field.