Semilocal right distributive skew Laurent series rings

We prove that the following conditions are equivalent. (1) The skew Laurent series ring $A((t,\varphi))$ is semilocal and right distributive. (2) The ring $A((t,\varphi))$ is a finite direct product of right uniserial rings. (3) The ring $A((t,\varphi))$ is a finite direct product of right uniserial right Artinian rings. (4) The ring $A$ is a finite direct product of right uniserial right Artinian rings $A_i$, and $\varphi(A_i)=A_i$ for all $i$.