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Study of skew inverse Laurent series rings. (English) Zbl 1392.16041

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

MSC:

16W60 Valuations, completions, formal power series and related constructions (associative rings and algebras)
16S99 Associative rings and algebras arising under various constructions
16P60 Chain conditions on annihilators and summands: Goldie-type conditions
16S36 Ordinary and skew polynomial rings and semigroup rings
16U80 Generalizations of commutativity (associative rings and algebras)
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