Abstract
We investigate the notions of weak annihilator and nilpotent associated prime defined by Ouyang and Birkenmeier (Weak annihilator over extension rings, 2012) in the setting of skew PBW extensions. 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, noncommutative algebraic geometry, and theoretical physics. Finally, we present some ideas for future research.