This chapter describes finitely presented algebras (FPAs) in Magma. An FPA is a quotient of a free associative algebra by an ideal of relations. To compute with these ideals, one constructs noncommutative Gröbner bases (GBs), which have many parallels with the standard commutative GBs, discussed in Chapter GRÖBNER BASES. At the heart of the theory is a noncommutative version of the Buchberger algorithm which computes a GB of an ideal of an algebra starting from an arbitrary basis (generating set) of the ideal. One significant difference with the commutative case is that a noncommutative GB may not be finite for a finitely-generated ideal. For overviews of the theory and the basic algorithms, see [Mor94], [Li02].
Magma also contains an implementation of a noncommutative generalization of the Faugere F4 algorithm (due to Allan Steel), based on sparse linear algebra techniques, which usually performs dramatically better than the Buchberger algorithm, and so this is used by Magma by default.