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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.