Let $\Lambda$ be a commutative ring with an automorphism $\sigma$ (I have in mind $\mathbb Z_p[[t]]$ and $\sigma(t) = (1+t)^\alpha - 1$ with $\alpha \in \Lambda^\times$) and $R = \Lambda\{F\}$ with $F\lambda = \sigma(\lambda)F$ for $\lambda \in \Lambda$.

Is there a classification of finitely generated modules over $R$? I am happy to assume that they are free (and finitely generated) as $\Lambda$ modules.

When $\Lambda$ is a field, there is a classification similar to the standard one over PID's in chapter three of "The theory of rings" by Nathan Jacobson.

What about the general case or at least my specific example? Or even when $\Lambda$ is a PID? Ideally, I would want any finitely generated module to be isomorphic to a direct sum of modules generated by one element, perhaps up to finite kernel and cokernel.