In the present note, we continue the study of skew inverse Laurent series ring R((x−1;α,δ))R((x−1;α,δ)) and skew inverse power series ring R[[x−1;α,δ]]R[[x−1;α,δ]], where RR is a ring equipped with an automorphism αα and an αα-derivation δδ. Necessary and sufficient conditions are obtained for R[[x−1;α,δ]]R[[x−1;α,δ]] to satisfy a certain ring property which is among being local, semilocal, semiperfect, semiregular, left quasi-duo, (uniquely) clean, exchange, projective-free and II-ring, respectively. It is shown here that R((x−1;α,δ))R((x−1;α,δ)) (respectively R[[x−1;α,δ]]R[[x−1;α,δ]]) is a domain satisfying the ascending chain condition (Acc) on principal left (respectively right) ideals if and only if so does RR. Also, we investigate the problem when a skew inverse Laurent series ring R((x−1;α,δ))R((x−1;α,δ)) has the same Goldie rank as the ring RR and is proved that, if RR is a semiprime right Goldie ring, then R((x−1;α,δ))R((x−1;α,δ)) is semiprimitive. Furthermore, we study on the relationship between the simplicity, semiprimeness, quasi-Baerness and Baerness property of a ring RR and these of the skew inverse Laurent series ring. Finally, we consider the problem of determining when f(x)∈R((x−1;α,δ))f(x)∈R((x−1;α,δ)) is nilpotent.
