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Let R be an arbitrary ring equipped with an automorphism α and an α-derivation δ. In this paper, we study on the relationship between the piecewise prime property of a ring R and these of the skew inverse Laurent series ring R((x−1; α,δ)) and related rings. Furthermore, for a piecewise prime ring, we determine a large class of rings which have a generalized triangular matrix representation for which the diagonal rings are prime. Our results extend and unify many available results. Examples for illustration and delimitation of the theory are provided. [ABSTRACT FROM AUTHOR]

AbstractLet R be a ring with a monomorphism α and an α-derivation δ. In this article, we give a simple and different proof about the semiprimitivity of Ore extensions which states that the skew polynomial ring R[x;α,δ] is semiprimitive reduced if and only if R is α-rigid. This unifies and extends a number of known results on the Jacobson radical in the special cases. Also, as an application of our results, by imposing constraints on α and δ, we completely identify the Jacobson radical of rings whose the set of all nilpotent elements has special conditions. Important examples are provided to illustrate the applications of the results. [ABSTRACT FROM AUTHOR]