This is inspired by the theorem mentioned in Why is this theorem attributed to J.-P. 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.

This is inspired by the theorem mentioned in Why is this theorem attributed to J.-P. 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.

This is inspired by the theorem mentioned in Why is this theorem attributed to J.-P. 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. $\lambda(M)=\text{lenght}(M)$ if $R$ is Artinian.

This is inspired by the theorem mentioned in Why is this theorem attributed to J.-P. 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.