  We investigate the notions of weak annihilator and nilpotent associated prime
defined by Ouyang and Birkenmeier \cite{OuyangBirkenmeier2012} 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.
