On Weak Annihilators and Nilpotent Associated Primes of Skew PBW Extensions

TL;DR

This paper extends the concepts of weak annihilators and nilpotent associated primes to noncommutative rings with PBW bases, broadening their application to various algebraic structures in mathematics and physics.

Contribution

It generalizes existing results on annihilators and associated primes from commutative and skew polynomial rings to PBW extensions, with applications in multiple advanced fields.

Findings

Extended the theory of annihilators and associated primes to PBW extensions.

Provided examples from enveloping algebras and noncommutative geometry.

Suggested directions for future research in noncommutative algebra.

Abstract

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.