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.