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In this paper, we study the reflexive-nilpotents-property (briefly, RNP) for Ore extensions of injective type, and more generally, skew PBW extensions. With this aim, we introduce the notions of Σ-skew CN rings and Σskew reflexive (RNP) rings, for Σ a finite family of ring endomorphisms of a ring R. Under certain conditions of compatibility, we study the transfer of the Σskew RNP property from a ring of coefficients to an Ore extension or skew PBW extension over this ring. We also consider this property for localizations of these noncommutative rings. Our results extend those corresponding presented by Bhattacharjee [9].

We investigate the notions of weak annihilator and nilpotent associated prime defined by Ouyang and Birkenmeier [82] in the setting of noncommutative rings having PBW bases. We extend several results formulated in the literature concerning annihilators and associated primes of commutative rings and skew polynomial rings to a more general setting of algebras not considered before. We exemplify our results with families of algebras appearing in the theory of enveloping algebras, differential operators on noncommutative spaces, noncommutative algebraic geometry, and theoretical physics. Finally, we present some ideas for future research.

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