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

  • an evolving toolkit for dealing with PBW-algebras and their quotients modulo two-sided ideals,
  • some 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

offer details on theory and algorithms.


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.