An affine algebra in Magma is simply the quotient ring of a multivariate polynomial ring P = R[x1, ..., xn] by an ideal J of P. Such rings arise commonly in commutative algebra and algebraic geometry. They can also be viewed as generalizations of number fields and algebraic function fields, when R is a field.
The elements of affine algebras are simply multivariate polynomials which are always kept reduced to normal form modulo the ideal J of "relations". Practically all operations which are applicable to multivariate polynomials are also applicable in Magma to elements of affine algebras (when meaningful).
If the ideal J of relations defining an affine algebra A = R[x1, ..., xn]/J is maximal and R is a field, then A is a field and may be used with any algorithms in Magma which work over fields. Factorization of polynomials over such affine algebras is also supported (including fields of small characteristic, since V2.10).
If an affine algebra defined over a field has finite dimension considered as a vector space over the coefficient field, extra special operations are available on its elements.
Currently the base ring R may be a field or a Euclidean ring. Further operations for affine algebras over Euclidean rings will be supported in the future.
An affine algebra has type RngMPolRes and its elements type RngMPolResElt.