On radicals of free skew extensions of a ring

Consider a set X of noncommuting variables, and let A={αx}x∈X and D={δx}x∈X be sequences of ring endomorphisms and skew derivations of a ring R. In this paper, we further investigate the free skew extension R[X;A,D] and comprehensively determine its Jacobson radical, given specific constraints on the coefficient ring. Additionally, we apply our findings to characterize when the free skew extension is 2-primal. Also, we provide partial characterizations for it to be NI. Moreover, we establish necessary and sufficient conditions for the free skew extension over a ring to exhibit certain properties, including being weak zip and weak symmetric. Consequently, we extend and unify several existing results.