# 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.
Note

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.

Note

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

offer details on theory and algorithms.