# Introduction

Working over a field $K$, our focus in this chapter is on noncommutative Gröbner bases and their application to the computational study of finitely presented associative $K$-algebras. At present state, OSCAR offers

- a comprehensive toolkit for dealing with PBW-algebras and their quotients modulo two-sided ideals,
- functionality for computing and applying (partial) two-sided Gröbner bases in free associative algebras on finitely many letters.

In contrast to the general case of finitely presented associative algebras, (left, right, two-sided) ideals in PBW-algebras admit finite (left, right, two-sided) Gröbner bases. In particular, PBW-algebras are Noetherian.

The class of PBW-algebras includes the Weyl algebras. Algebras which arise as quotients of PBW-algebras include the Clifford algebras (in particular, the exterior algebras).

The textbooks

and the thesis

- [Lev05]

offer details on theory and algorithms.

## Contact

Please direct questions about this part of OSCAR to the following people:

You can ask questions in the OSCAR Slack.

Alternatively, you can raise an issue on github.