# Finitely Presented Algebras

A finitely presented algebra (fp-algebra) in Magma is simply the quotient ring of a free algebra $F = R\langle x_1,\ldots,x_n\rangle$ by an ideal $J$ of $F$. It is an object of type `AlgFP` with elements of type `AlgFPElt`.

The elements of fp-algebras are simply noncommutative polynomials which are always kept reduced to normal form modulo the ideal $J$ of “relations”. Practically all operations which are applicable to noncommutative polynomials are also applicable in Magma to elements of fp-algebras (when meaningful).

If an fp-algebra $A$ has finite dimension, considered as a vector space over its coefficient field, then extra special operations are available for $A$ and its elements.