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Macaulay2 web site
Ring -- the class of all rings
Description
Common ways to make a ring:
Ring / Ideal
-- make a quotient ring
Ring Array
-- the standard way to make a polynomial ring
GF
-- make a finite field
Common functions for accessing the variables or elements in a ring:
use(Ring)
-- install ring variables and ring operations
generators(Ring)
-- the list of generators of a ring
numgens(Ring)
-- number of generators of a polynomial ring
Ring _ ZZ
-- get a ring variable by index
Common ways to get information about a ring:
char(Ring)
-- computes the characteristic of the ring or field
coefficientRing(Ring)
-- get the coefficient ring
dim(Ring)
-- compute the Krull dimension
Common ways to use a ring:
Ring ^ ZZ
-- make a free module
Ring ^ List
-- make a free module
vars(Ring)
-- row matrix of the variables
See also
rings
Types of ring :
EngineRing
-- the class of rings handled by the engine
Functions and methods returning a ring :
ambient(GaloisField)
-- corresponding quotient ring
ambient(QuotientRing), see
ambient(Ring)
-- ambient polynomial ring
ambient(Ring)
-- ambient polynomial ring
associatedGradedRing
-- the associated graded ring of an ideal
coefficientRing(Ring), see
coefficientRing
-- get the coefficient ring
integralClosure(Ring)
-- compute the integral closure (normalization) of an affine domain
minimalPresentation(Ring)
-- compute a minimal presentation of a quotient ring
prune(Ring), see
minimalPresentation(Ring)
-- compute a minimal presentation of a quotient ring
newRing
-- make a copy of a ring, with some features changed
normalCone
-- the normal cone of a subscheme
reesAlgebra
-- compute the defining ideal of the Rees Algebra
ring
-- get the associated ring of an object
Ring ** Ring
-- tensor product
coefficientRing(SchurRing), see
SchurRing
-- the class of all Schur rings
specialFiber
-- special fiber of a blowup
symmetricAlgebra(Module), see
symmetricAlgebra
-- the symmetric algebra of a module
tensor(Ring,Ring)
-- tensor product
trim(Ring)
Methods that use a ring :
Ideal * Ring, see
*
-- a binary operator, usually used for multiplication
MonomialIdeal * Ring, see
*
-- a binary operator, usually used for multiplication
Ring * Ideal, see
*
-- a binary operator, usually used for multiplication
Ring * MonomialIdeal, see
*
-- a binary operator, usually used for multiplication
Ring * RingElement, see
*
-- a binary operator, usually used for multiplication
Ring * Vector, see
*
-- a binary operator, usually used for multiplication
Ideal == Ring, see
==
-- equality
MonomialIdeal == Ring, see
==
-- equality
Ring == Ideal, see
==
-- equality
Ring == MonomialIdeal, see
==
-- equality
Ring == ZZ, see
==
-- equality
ZZ == Ring, see
==
-- equality
AffineVariety ** Ring
-- a binary operator, usually used for tensor product or Cartesian product
basis(InfiniteNumber,InfiniteNumber,Ring), see
basis
-- basis of all or part of a module or ring
basis(InfiniteNumber,List,Ring), see
basis
-- basis of all or part of a module or ring
basis(InfiniteNumber,ZZ,Ring), see
basis
-- basis of all or part of a module or ring
basis(List,InfiniteNumber,Ring), see
basis
-- basis of all or part of a module or ring
basis(List,List,Ring), see
basis
-- basis of all or part of a module or ring
basis(List,Ring), see
basis
-- basis of all or part of a module or ring
basis(List,ZZ,Ring), see
basis
-- basis of all or part of a module or ring
basis(Ring), see
basis
-- basis of all or part of a module or ring
basis(ZZ,InfiniteNumber,Ring), see
basis
-- basis of all or part of a module or ring
basis(ZZ,List,Ring), see
basis
-- basis of all or part of a module or ring
basis(ZZ,Ring), see
basis
-- basis of all or part of a module or ring
basis(ZZ,ZZ,Ring), see
basis
-- basis of all or part of a module or ring
ChainComplex ** Ring
-- a binary operator, usually used for tensor product or Cartesian product
chainComplex(Ring)
-- make an empty chain complex over a ring
char(Ring), see
char
-- computes the characteristic of the ring or field
conductor(Ring), see
conductor
-- the conductor of a finite ring map
degree(Ring)
degreeLength(Ring), see
degreeLength
-- the number of degrees
degrees(Ring)
-- degrees of generators
degreesRing(Ring)
-- the ring of degrees
diagonalMatrix(Ring,List), see
diagonalMatrix(Ring,ZZ,ZZ,List)
-- make a diagonal matrix from a list
diagonalMatrix(Ring,ZZ,ZZ,List)
-- make a diagonal matrix from a list
dim(Ring)
-- compute the Krull dimension
euler(Ring)
-- Euler characteristic
eulers(Ring)
-- list the sectional Euler characteristics
Ext(Ideal,Ring), see
Ext(Module,Module)
-- total Ext module
Ext(Module,Ring), see
Ext(Module,Module)
-- total Ext module
Ext^ZZ(Matrix,Ring), see
Ext^ZZ(Matrix,Module)
-- map between Ext modules
Ext^ZZ(Ideal,Ring), see
Ext^ZZ(Module,Module)
-- Ext module
Ext^ZZ(Module,Ring), see
Ext^ZZ(Module,Module)
-- Ext module
Fano(ZZ,Ideal,Ring)
-- Fano scheme
flattenRing(Ring), see
flattenRing
-- write a ring as a (quotient) of a polynomial ring over ZZ or a prime field
frac(Ring), see
frac
-- construct a fraction field
genera(Ring)
-- list of the successive linear sectional arithmetic genera
generators(Ring)
-- the list of generators of a ring
genericMatrix(Ring,RingElement,ZZ,ZZ), see
genericMatrix
-- make a generic matrix of variables
genericMatrix(Ring,ZZ,ZZ), see
genericMatrix
-- make a generic matrix of variables
genericSkewMatrix(Ring,RingElement,ZZ), see
genericSkewMatrix
-- make a generic skew symmetric matrix of variables
genericSkewMatrix(Ring,ZZ), see
genericSkewMatrix
-- make a generic skew symmetric matrix of variables
genericSymmetricMatrix(Ring,RingElement,ZZ), see
genericSymmetricMatrix
-- make a generic symmetric matrix
genericSymmetricMatrix(Ring,ZZ), see
genericSymmetricMatrix
-- make a generic symmetric matrix
genus(Ring)
-- arithmetic genus
GF(Ring), see
GF
-- make a finite field
heft(Ring), see
heft
-- heft vector of ring, module, graded module, or resolution
hilbertFunction(List,Ring), see
hilbertFunction
-- the Hilbert function
hilbertFunction(ZZ,Ring), see
hilbertFunction
-- the Hilbert function
hilbertPolynomial(Ring)
-- compute the Hilbert polynomial of the ring
Hom(Ideal,Ring), see
Hom(Module,Module)
-- module of homomorphisms
Hom(Module,Ring), see
Hom(Module,Module)
-- module of homomorphisms
Hom(Ring,Ideal), see
Hom(Module,Module)
-- module of homomorphisms
Hom(Ring,Module), see
Hom(Module,Module)
-- module of homomorphisms
icFracP(Ring), see
icFracP
-- compute the integral closure in prime characteristic
icFractions(Ring), see
icFractions
-- fractions integral over an affine domain
icMap(Ring), see
icMap
-- natural map from an affine domain into its integral closure
ideal(Ring)
-- returns the defining ideal
IndexedVariable _ Ring
-- get a ring variable by name
isAffineRing(Ring), see
isAffineRing
-- whether something is an affine ring
isCommutative(Ring), see
isCommutative
-- whether a ring is commutative
isField(Ring), see
isField
-- whether something is a field
isHomogeneous(Ring), see
isHomogeneous
-- whether something is homogeneous (graded)
isNormal(Ring), see
isNormal
-- determine if a reduced ring is normal
isQuotientOf(Ring,QuotientRing), see
isQuotientOf(Ring,Ring)
-- whether one ring is a quotient of another
isQuotientOf(Ring,Ring)
-- whether one ring is a quotient of another
isQuotientOf(Type,Ring)
-- whether one ring is a quotient of a ring of a given type
isQuotientRing(Ring), see
isQuotientRing
-- whether something is a quotient ring
isRing(Ring), see
isRing
-- whether something is a ring
isSkewCommutative(Ring), see
isSkewCommutative
-- whether a ring has skew commuting variables
jacobian(Ring)
-- the Jacobian matrix of the polynomials defining a quotient ring
Constant ^ Ring, see
lift
-- lift to another ring
Number ^ Ring, see
lift
-- lift to another ring
RingElement ^ Ring, see
lift
-- lift to another ring
makeS2(Ring), see
makeS2
-- compute the S2ification of a reduced ring
map(Ring,Matrix)
-- make a ring map
map(Ring,Ring)
-- make a ring map, using the names of the variables
map(Ring,Ring,List)
-- make a ring map
map(Ring,Ring,Matrix)
-- make a ring map
map(Ring,Ring,RingMap), see
map(Ring,Ring,Matrix)
-- make a ring map
Matrix ** Ring
-- tensor product
Ring ** Matrix, see
Matrix ** Ring
-- tensor product
matrix(Ring,List)
-- create a matrix from a doubly nested list of ring elements or matrices
Module ** Ring
-- tensor product
Ring ** Module, see
Module ** Ring
-- tensor product
module(Ring)
multidegree(Ring), see
multidegree
-- multidegree
mutableIdentity(Ring,ZZ)
-- make a mutable identity matrix
mutableMatrix(Ring,ZZ,ZZ)
-- make a mutable matrix filled with zeroes
numgens(Ring)
-- number of generators of a polynomial ring
options(Ring)
-- get values used for optional arguments
poincare(Ring)
-- assemble degrees of an ring into a polynomial
precision(Ring), see
precision
Proj(Ring)
-- make a projective variety
Number _ Ring, see
promote
-- promote to another ring
RingElement _ Ring, see
promote
-- promote to another ring
random(List,Ring), see
random(Type)
-- random element of a type
random(ZZ,Ring), see
random(Type)
-- random element of a type
Ring / Ideal
-- make a quotient ring
Ring / List, see
Ring / Ideal
-- make a quotient ring
Ring / Module, see
Ring / Ideal
-- make a quotient ring
Ring / MonomialIdeal, see
Ring / Ideal
-- make a quotient ring
Ring / RingElement, see
Ring / Ideal
-- make a quotient ring
Ring / Sequence, see
Ring / Ideal
-- make a quotient ring
Ring / ZZ, see
Ring / Ideal
-- make a quotient ring
Ring ^ List
-- make a free module
Ring ^ ZZ
-- make a free module
Ring _ List
-- make a monomial from a list of exponents
Ring _ String
-- get a ring variable by name
Ring _ ZZ
-- get a ring variable by index
Ring _*
(missing documentation)
Ring Array
-- the standard way to make a polynomial ring
Ring List
-- make a local polynomial ring
Ring OrderedMonoid
-- make a polynomial ring
Ring ~, see
sheaf(Ring)
-- make a coherent sheaf of rings
sheaf(Ring)
-- make a coherent sheaf of rings
sheaf(Variety,Ring)
-- make a coherent sheaf of rings
singularLocus(Ring), see
singularLocus
-- singular locus
Spec(Ring)
-- make an affine variety
substitute(Ideal,Ring), see
substitute
-- substituting values for variables
substitute(Matrix,Ring), see
substitute
-- substituting values for variables
substitute(Module,Ring), see
substitute
-- substituting values for variables
substitute(Number,Ring), see
substitute
-- substituting values for variables
substitute(RingElement,Ring), see
substitute
-- substituting values for variables
substitute(Vector,Ring), see
substitute
-- substituting values for variables
Symbol _ Ring
-- get a ring variable by name
symmetricAlgebra(Nothing,Ring,Matrix), see
symmetricAlgebra
-- the symmetric algebra of a module
symmetricAlgebra(Ring,Nothing,Matrix), see
symmetricAlgebra
-- the symmetric algebra of a module
symmetricAlgebra(Ring,Ring,Matrix), see
symmetricAlgebra
-- the symmetric algebra of a module
tensor(Ring,RingMap,Matrix)
-- tensor product via a ring map
tensor(Ring,RingMap,Module), see
tensor(Ring,RingMap,Matrix)
-- tensor product via a ring map
terms(Ring,RingElement), see
terms
-- provide a list of terms of a polynomial
toField(Ring)
-- declare that a ring is a field
use(Ring)
-- install ring variables and ring operations
vars(Ring)
-- row matrix of the variables
Fixed objects of class Ring :
QQ
-- the class of all rational numbers
ZZ
-- the class of all integers
For the programmer
The object
Ring
is
a
type
, with ancestor classes
Type
<
MutableHashTable
<
HashTable
<
Thing
.