Modules over Affine Algebras

Modules over a multivariate polynomial ring $R[x_1, \ldots, x_n]$(R a Euclidean ring or field) and quotient rings of such (affine algebras) form a special category in Magma. Multivariate polynomial rings are not principal ideal rings in general, so the standard matrix echelonization algorithms are not applicable. Magma allows computations in modules over such rings by adding a column field to each monomial of a polynomial and then by using the ideal machinery based on Gröbner bases. This method is much more efficient than introducing new variables to represent the columns since the number of columns does not affect the total number of variables.