Real-valued dimension

Let $\overline{\mathbb{R}}_{\geq 0} = \mathbb{R}_{\geq 0} \cup \{\infty\}$.

Does there exist an example of the following?

• A commutative ring with unity $R$
• A mapping $\operatorname{d}: R\operatorname{-Mod} \to \overline{\mathbb{R}}_{\geq 0}$ with the following properties:
  • If $A \cong B$, $\operatorname{d}(A) = \operatorname{d}(B)$
  • $\operatorname{d}(\{0\}) = 0$
  • $\operatorname{d}(R) = 1$
  • $\operatorname{d}(A \oplus B) = \operatorname{d}(A) + \operatorname{d}(B)$
  • $\operatorname{d}(A \otimes B) = \operatorname{d}(A) \cdot \operatorname{d}(B)$
  • If $A \subseteq B$ is a submodule, then $\operatorname{d}(A) \leq \operatorname{d}(B)$
  • If $A \subseteq B$ is a submodule, then $\operatorname{d}(B/A) \leq \operatorname{d}(B)$
  • If $A \subseteq B$ is a submodule, then $\operatorname{d}(A) + \operatorname{d}(B/A) = \operatorname{d}(B)$
  • $\operatorname{d}$ is surjective