The Jacobson radical of the Laurent series ring

For a large class of rings $A$ including all rings with right Krull dimension, it is proved that for every automorphism $\varphi$ of the ring $A$, the Jacobson radical of the skew Laurent series ring $A((x,\varphi))$ is nilpotent and coincides with $N((x,\varphi))$, where $N$ is the prime radical of the ring $A$. If $A/N$ is a ring of bounded index, then the Jacobson radical of the Laurent series ring $A((x))$ coincides with $N((x))$.