Ring -- the class of all rings

Ring -- the class of all rings

The type Ring is a member of the class Type. Each object of class Ring is called a ring. Each ring is also a member of class Type.

More general types (whose methods may also apply) :

Type -- the class of all types

MutableHashTable -- the class of all mutable hash tables

HashTable -- the class of all hash tables

Types of ring :

EngineRing -- the class of rings handled by the engine

See also:

rings

Common ways to make a ring:

Ring / Ideal -- quotient ring

Ring [...] -- the standard way to make a polynomial ring

GF -- make a finite field

Common functions for accessing the variables or elemenets in a ring:

use Ring -- install defaults

generators Ring -- matrix of generators

numgens Ring -- the number of generators

Ring _ ZZ -- get a variable by number

ZZ _ Ring

Common ways to get information about a ring:

char Ring -- the characteristic of a field or ring

coefficientRing Ring -- get the coefficient ring

dim Ring -- calculate the dimension

Common ways to use a ring:

Ring ^ ZZ -- make a free module

vars Ring -- row matrix of the variables

Functions and methods returning a a ring :

ambient GaloisField

ambient QuotientRing

ambient Ring

coefficientRing GaloisField

coefficientRing PolynomialRing

coefficientRing Ring

integralClosure Ring -- compute the integral closure of a ring

modifyRing Ring

Ring ** Ring -- tensor product

ring -- get the associated ring

tensor(Ring,Ring)

trim Ring

Methods for using a ring :

AffineVariety ** Ring

allGenerators Ring

basis Ring -- basis of a ring

basis(List,Ring)

basis(ZZ,Ring)

char Ring

degree Ring

degreeLength Ring

degrees Ring -- degrees of generators

dim Ring

euler Ring

Ext(Ideal,Ring)

Ext(Module,Ring)

Ext^ZZ(Ideal,Ring)

Ext^ZZ(Matrix,Ring)

Ext^ZZ(Module,Ring)

Fano(ZZ,Ideal,Ring)

GF Ring -- make a finite field from a ring

Grassmannian(ZZ,ZZ,Ring)

group Ring

hilbertFunction(List,Ring)

hilbertFunction(ZZ,Ring)

hilbertPolynomial Ring

Hom(Ideal,Ring)

Hom(Module,Ring)

Hom(Ring,Ideal)

Hom(Ring,Module)

ICfractions Ring -- Compute the fractions integral over a domain.

ICfractionsLong Ring

ICmap Ring -- natural map from an affine domain into its integral closure.

Ideal * Ring

Ideal == Ring

ideal Ring -- get the ideal used to form a quotient ring

IndexedVariable _ Ring -- generator of a ring with a given name

isNormal Ring -- determine if a reduced ring is normal

isQuotientOf(QuotientRing,Ring)

isQuotientOf(Ring,Ring)

isQuotientRing Ring

isRing Ring

jacobian Ring

lift(Ideal,Ring)

lift(Matrix,Ring)

lift(QQ,Ring)

lift(RingElement,Ring)

lift(ZZ,Ring)

liftable(QQ,Ring)

liftable(RingElement,Ring)

liftable(ZZ,Ring)

map(Ring,Matrix)

map(Ring,Ring)

map(Ring,Ring,{...}) -- make a ring map

map(Ring,Ring,Matrix) -- make a ring map

Matrix ** Ring -- tensor product

matrix(Ring,{...}) -- make a matrix

minPres Ring

minPresMap Ring

Module ** Ring -- tensor product

module Ring

monoid Ring -- get the monoid from a monoid ring

MonomialIdeal * Ring

MonomialIdeal == Ring

monomialIdeal Ring

new EngineRing from Ring

new Module from Ring

numgens Ring

options Ring -- get values used for optional arguments

poincare Ring

Proj Ring

ProjectiveVariety ** Ring

promote(MonoidElement,Ring)

promote(QQ,Ring)

promote(RingElement,Ring)

promote(ZZ,Ring)

QQ _ Ring

random Ring -- random element of a ring

random({...},Ring) -- a random ring element of a given degree

random(ZZ,Ring) -- a random ring element of a given degree

Ring * Ideal

Ring * MonomialIdeal

Ring / (...) -- quotient ring

Ring / List

Ring / Ideal -- quotient ring

Ring / Module

Ring / MonomialIdeal

Ring / RingElement -- quotient ring

Ring / ZZ

Ring == Ideal

Ring == MonomialIdeal

Ring == ZZ

Ring [...] -- the standard way to make a polynomial ring

Ring ^ {...} -- make a free module

Ring ^ ZZ -- make a free module

Ring _ {...} -- make a monomial from a list of exponents

Ring _ String -- get a variable by name

Ring _ ZZ -- get a variable by number

Ring ~ -- make the structure sheaf

Ring OrderedMonoid -- make a polynomial ring

RingElement _ Ring -- generator of a ring with a given name

sheaf Ring -- make the structure sheaf

sheaf(Variety,Ring)

singularLocus Ring

Spec Ring

substitute(Ideal,Ring)

substitute(Matrix,Ring)

substitute(Module,Ring)

substitute(RingElement,Ring)

substitute(Vector,Ring)

Symbol _ Ring -- generator of a ring with a given name

use Ring

vars Ring -- row matrix of the variables

ZZ == Ring

ZZ _ Ring

Fixed objects of class Ring :

CC -- the class of all complex numbers

QQ -- the class of all rational numbers

RR -- the class of all real numbers

ZZ -- the class of all integers