This chapter describes the Magma functionality for ideals over polynomial rings. For the basics on multivariate polynomial rings and their elements, see Chapter MULTIVARIATE POLYNOMIAL RINGS. Most of the significant operations with ideals construct or utilise a previously-constructed Gröbner basis. The monomial ordering used for this basis can greatly affect the speed and memory usage of these operations. This ordering is attached to the polynomial ring in which the ideals are created. For information on Gröbner bases and the creation of polynomial rings with specified orders, see Chapter GRÖBNER BASES. That chapter also tells the user how to compute and return a Gröbner basis, or just to compute it internally for later use in the operations described below, with many additional configuration parameters to optimise the computation. Users may ignore the issue when creating the ambient polynomial rings by allowing Magma to make default choices. It is, however, highly recommended that users who wish to work with complicated ideals thoroughly acquaint themselves with the options available. Magma has an extremely powerful Gröbner basis engine and often makes sophisticated choices internally of alternative monomial orders for particular computations. Ultimately, however, the user may significantly speed up his work by a judicious choice of order. We note here that the default order is the lexicographical one, a total elimination order well suited to finding solutions of zero-dimensional systems of polynomial equations but tending to produce very large bases that can take much time and memory to compute. For homogeneous ideals of rings with the standard weighting (all variables have weight one), the grevlex order is usually the best in practice and there is theoretical justification for this. In the case that the ring has a different weighting and the ideal is homogeneous with respect to that, the weighted grevlex order is the best choice. In any case, the EasyIdeal and EasyBasis intrinsics of the Gröbner basis chapter return to the user a basis for an internally chosen good order and these "easy" bases are used in many internal functions if a basis with respect to the polynomial ring order has not already been computed and stored.
The functions and operations described here cover a wide range of commutative algebra functionality. This includes sums and intersections, colon ideals and saturations, elimination, radicals and primary decompositions, Noether normalisations and computation of Hilbert polynomials and Hilbert series.
Related chapters including other polynomial ring functionality relying on Gröbner bases are the chapter on invariant rings of finite group actions, Chapter INVARIANT THEORY, and the chapters on affine algebras (Chapter AFFINE ALGEBRAS) and on modules over affine algebras (Chapter MODULES OVER MULTIVARIATE RINGS). The chapter on algebraically closed fields (Chapter ALGEBRAICALLY CLOSED FIELDS) describes functions that allows one to compute the variety of an ideal over the algebraic closure of the base field. And, of course, the Algebraic Geometry component of Magma and parts of the Arithmetic Geometry are built upon the commutative algebra here.