# Introduction

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](../../CommutativeAlgebra/GrobnerBases/index-grobner-bases.md#chapgb). 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 [[Li, 2002](../../references.md#cite-li-noncommut), [Mora, 1994](../../references.md#cite-moranoncommut)]. Magma also contains an implementation of a noncommutative generalization of the Faugere $F_4$ 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.