Ring -- the class of all rings

Ring -- the class of all rings

A ring is a set together with operations +, -, * and elements 0, 1 satisfying the usual rules. In this system, it is also understood to be a ZZ-algebra, which means that the operations where one argument is an integer are also provided.

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.

Types of Ring :

EngineRing -- the class of rings handled by the engine

Field -- the class of all fields

ProductRing -- the class of all product rings

Making a ring :

Ring ** Ring -- tensor product

ambient GaloisField

ambient QuotientRing

ambient Ring

coefficientRing GaloisField

coefficientRing PolynomialRing

coefficientRing Ring

modifyRing Ring

ring -- get the associated ring

tensor(Ring,Ring)

trim Ring

Methods for using a ring :

AffineVariety ** Ring

Ext(ZZ,Ideal,Ring)

Ext(ZZ,Matrix,Ring)

Ext(ZZ,Module,Ring)

Ext(ZZ,Ring,Ideal)

Ext(ZZ,Ring,Module)

Ext(ZZ,Ring,Ring)

Fano(ZZ,Ideal,Ring)

GF Ring -- make a finite field from a ring

Grassmannian(ZZ,ZZ,Ring)

Hom(Ideal,Ring)

Hom(Module,Ring)

Hom(Ring,Ideal)

Hom(Ring,Module)

Ideal * Ring

Ideal == Ring

Matrix ** Ring -- tensor product

Module ** Ring -- tensor product

Proj Ring

ProjectiveVariety ** Ring

Ring * Ideal

Ring * Ring

Ring / (...)

Ring / Ideal -- quotient ring

Ring / Module

Ring / RingElement

Ring / ZZ

Ring / {...}

Ring == Ideal

Ring OrderedMonoid -- make a polynomial ring

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

Ring ^ ZZ -- make a free module

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

Ring _ String -- get a variable by name

Ring _ ZZ -- get a variable by number

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

Spec Ring

ZZ _ Ring

basis Ring

basis(ZZ,Ring)

basis({...},Ring)

char Ring

degreeLength Ring

dim Ring

euler Ring

frac Ring

genera Ring

generators Ring

group Ring

hilbertFunction(ZZ,Ring)

hilbertFunction({...},Ring)

hilbertPolynomial Ring

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

isAffineRing Ring

isCommutative Ring

isField Ring

isHomogeneous Ring

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,Matrix)

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

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

monoid Ring -- get the monoid from a monoid ring

new EngineRing from Ring

new Module from Ring

numgens Ring

options Ring -- get values used for optional arguments

poincare Ring

promote(MonoidElement,Ring)

promote(QQ,Ring)

promote(RingElement,Ring)

promote(ZZ,Ring)

random Ring -- random element of a ring

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

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

sheaf Ring -- make the structure sheaf

singularLocus Ring

substitute(Ideal,Ring)

substitute(Matrix,Ring)

substitute(Module,Ring)

substitute(RingElement,Ring)

substitute(Vector,Ring)

use Ring

vars Ring

Fixed objects of class Ring :

ZZ -- the class of all integers