The Magma facility for commutative rings allows the user to define any ring, starting from the ring of integers, by repeatedly applying the four basic constructions: transcendental extension, quotient by an ideal, localization, and completion. Rings derived from a polynomial ring will be considered in this section, while fields, their orders and valuation rings will be presented in the following section. The following rings and modules are considered here:
The basic computational problems for commutative rings include:
The fundamental tools on which most machinery for computational (commutative) ring theory is based include factorization of elements in a UFD, the efficient construction of standard bases for ideals and the factorization of ideals.