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4.8.1 Introduction to Rings

Polynomial rings play a central role in CoCoA. Indeed, every object in CoCoA is defined over a base ring which is a polynomial ring. The user can define many rings, but at any time a current ring is active within the system. Most commands use the current ring as the base ring.

Once a ring has been defined, the system can handle the following mathematical objects defined over that ring:

 * numbers (integers, rationals, modular integers);
 * polynomials;
 * vectors of polynomials;
 * rational functions;
 * ideals;
 * modules (submodules of a free module);
 * lists of objects;
 * matrices of objects.

Variables containing ring-dependent objects such as polynomials, ideals, and modules are labeled by their ring. Variables containing objects such as integers which are not dependent on a particular ring are not labeled.

IMPORTANT NOTE: Starting with CoCoA 3.5, variables are no longer local to specific rings, i.e., all variables are accessible from all rings.

The next sections explains how to create a ring.