It is shown that the first-order theory Th R (A) of a countable module over an arbitrary countable ring R is ℵ 0 -categorical if and only if $A \cong \bigoplus_{t finite, n ∈ ω, κ i ≤ ω. Furthermore, Th R (A) is ℵ 0 -categorical for all R-modules A if and only if R is finite and there exist only finitely many isomorphism classes of indecomposable R-modules.

It is shown that the first-order theory $\mathrm{Th}_R(A)$ of a countable module over an arbitrary countable ring $R$ is $\aleph_0$-categorical if and only if $A \cong \bigoplus_{t < n}A_i^{(\kappa_i)}, A_i$ finite, $n \in \omega, \kappa_i \leq \omega$. Furthermore, $\mathrm{Th}_R(A)$ is $\aleph_0$-categorical for all $R$-modules $A$ if and only if $R$ is finite and there exist only finitely many isomorphism classes of indecomposable $R$-modules.