Finitely-presented associative algebras (non-commutative polynomial rings) are defined by taking R-linear combinations of elements of a semigroup or group, where R is some ring.
There are two major tools for computing with these algebras. Linton's vector enumerator uses the Todd-Coxeter algorithm in an attempt to construct a matrix representation. If the user has some idea as to how to select ideals that might give rise to matrix representations of reasonable degree, this approach is very successful. The other approach is to apply a version of Buchberger's algorithm to construct a Gröbner basis for an ideal. This technique has been developed chiefly by Teo Mora in Genova and Ed Green in Virginia.