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A finitely presented algebra (fp-algebra) in Magma is simply
the quotient ring of a free algebra F = R< x1, ..., xn >
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.
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