Von Neumann regular skew-Laurent series rings

Let $\varphi$ be an automorphism of finite order. Then the skew-Laurent series ring $A((x,\varphi))$ is von Neumann regular iff $A$ is semisimple Artinian. The third equivalent condition is that $A((x,\varphi))$ is semisimple Artinian. The same result for strong regularity is proved in the case of an arbitrary automorphism $\varphi$.