An alternative perspective on Jacobson radical of skew inverse Laurent series rings

Abstract

In this paper, we continue to study skew inverse Laurent series ring [Formula: see text], where [Formula: see text] is a ring equipped with an automorphism [Formula: see text] and an [Formula: see text]-derivation [Formula: see text]. We directly prove that [Formula: see text] is semiprimitive reduced if and only if [Formula: see text] is [Formula: see text]-rigid. Also, as an application of our results, by imposing constraints on [Formula: see text] and [Formula: see text], we completely identify the Jacobson radical of [Formula: see text] 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.