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In this paper, we continue to study skew inverse Laurent series ring R((x−1; α,δ)), where R is a ring equipped with an automorphism α and an α-derivation δ. We directly prove that R((x−1; α,δ)) is semiprimitive reduced if and only if R is α-rigid. Also, as an application of our results, by imposing constraints on α and δ, we completely identify the Jacobson radical of R((x−1; α,δ)) whose set of all nilpotent elements has special conditions. Moreover, necessary and sufficient conditions are obtained for the skew inverse Laurent series ring to satisfy a certain ring property which is among being right Artinian, completely primary, right perfect, (semi)local, semiperfect, semiprimary, semiregular, semisimple and strongly regular, respectively. [ABSTRACT FROM AUTHOR]

Let R be an associative ring equipped with an automorphism α and an α -derivation δ. In this note, we characterize when a skew inverse Laurent series ring R ((x − 1 ; α , δ)) and a skew inverse power series ring R [ [ x − 1 ; α , δ ] ] are 2-primal, and we obtain partial characterizations for those to be NI. [ABSTRACT FROM AUTHOR]

The aim of our paper is to study the primitivity of the skew inverse Laurent series rings R ((x − 1 ; α , δ)) and the skew Laurent power series rings R [ [ x , x − 1 ; α ] ] , where R is an associative ring equipped with an automorphism α and an α -derivation δ. Examples to illustrate and delimit the theory are provided. [ABSTRACT FROM AUTHOR]

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