Narrow Results

In this paper, we continue to study the differential inverse power series ring $R[[x^{-1};\delta ]]$, where $R$ is a ring equipped with a derivation $\delta$. We characterize when $R[[x^{-1};\delta ]]$ is a local, semilocal, semiperfect, semiregular, left quasi-duo, (uniquely) clean, exchange, right stable range one, abelian, projective-free, $I$-ring, respectively. Furthermore, we prove that $R[[x^{-1};\delta ]]$ is a domain satisfying the $ACC$ on principal left ideals if and only if so does $R$. Also, for a piecewise prime ring (PWP) $R$ we determine a large class of the differential inverse power series ring $R[[x^{-1};\delta ]]$ which have a generalized triangular matrix representation for which the diagonal rings are prime. In particular, it is proved that, under suitable conditions, if $R$ has a (flat) projective socle, then so does $R[[x^{-1};\delta ]]$. Our results extend and unify many existing results.

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.