# 5 Rings

Rings are important algebraic domains. Mathematically a ring is a set R with two operations `+` and `*` called addition and multiplication. (R,+) must be an abelian group. The identity of this group is called 0_R. (R-{0_R},*) must be a monoid. If this monoid has an identity element it is called 1_R.

Important examples of rings are the integers (see Integers), the Gaussian integers (see Gaussians), the integers of a cyclotomic field (see Subfields of Cyclotomic Fields), and matrices (see Matrices).

This chapter contains sections that describe how to test whether a domain is a ring (see IsRing), and how to find the smallest and the default ring in which a list of elements lies (see Ring and DefaultRing).

The next sections describe the operations applicable to ring elements (see Comparisons of Ring Elements, Operations for Ring Elements, Quotient).

The next sections describe the functions that test whether a ring has certain properties (IsCommutativeRing, IsIntegralRing, IsUniqueFactorizationRing, and IsEuclideanRing).

The next sections describe functions that are related to the units of a ring (see IsUnit, Units, IsAssociated, StandardAssociate, and Associates).

Then come the sections that describe the functions that deal with the irreducible and prime elements of a ring (see IsIrreducible, IsPrime, and Factors).

Then come the sections that describe the functions that are applicable to elements of rings (see EuclideanDegree, EuclideanRemainder, EuclideanQuotient, QuotientRemainder, QuotientMod, PowerMod, Gcd, GcdRepresentation, Lcm).

The last section describes how ring records are represented internally (see Ring Records).

Because rings are a category of domains all functions applicable to domains are also applicable to rings (see chapter Domains) .

All functions described in this chapter are in `LIBNAME/"ring.g"`.

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GAP 3.4.4
April 1997