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# 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:
Common ways to get information about a ring:
Common ways to use a ring:

## Types of ring :

• EngineRing -- the class of rings handled by the engine

## 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
• associatedPrimes(Ring) -- see associatedPrimes -- find associated primes
• baseRing(Ring) -- see baseRing -- produce the ring from which a ring was formed
• basis(InfiniteNumber,InfiniteNumber,Ring) -- see basis -- basis or generating set of all or part of a ring, ideal or module
• basis(InfiniteNumber,List,Ring) -- see basis -- basis or generating set of all or part of a ring, ideal or module
• basis(InfiniteNumber,ZZ,Ring) -- see basis -- basis or generating set of all or part of a ring, ideal or module
• basis(List,InfiniteNumber,Ring) -- see basis -- basis or generating set of all or part of a ring, ideal or module
• basis(List,List,Ring) -- see basis -- basis or generating set of all or part of a ring, ideal or module
• basis(List,Ring) -- see basis -- basis or generating set of all or part of a ring, ideal or module
• basis(List,ZZ,Ring) -- see basis -- basis or generating set of all or part of a ring, ideal or module
• basis(Ring) -- see basis -- basis or generating set of all or part of a ring, ideal or module
• basis(ZZ,InfiniteNumber,Ring) -- see basis -- basis or generating set of all or part of a ring, ideal or module
• basis(ZZ,List,Ring) -- see basis -- basis or generating set of all or part of a ring, ideal or module
• basis(ZZ,Ring) -- see basis -- basis or generating set of all or part of a ring, ideal or module
• basis(ZZ,ZZ,Ring) -- see basis -- basis or generating set of all or part of a ring, ideal or module
• Ring ^ BettiTally -- see BettiTally -- the class of all Betti tallies
• 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
• coefficientRing(Ring) -- see coefficientRing -- get the coefficient ring
• conductor(Ring) -- see conductor -- the conductor of a finite ring map
• degree(Ring)
• degreeGroup(Ring) (missing documentation)
• degreeLength(Ring) -- see degreeLength -- the length of the degree vector
• degrees(Ring) -- degrees of generators
• degreesMonoid(Ring) -- see degreesRing -- the ring or monoid of degrees
• degreesRing(Ring) -- see degreesRing -- the ring or monoid 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
• effCone(Ring) (missing documentation)
• effGenerators(Ring) (missing documentation)
• euler(Ring)
• eulers(Ring)
• Ext(Ideal,Ring) -- see Ext(Module,Module) -- total Ext module
• Ext(Module,Ring) -- see Ext(Module,Module) -- total Ext module
• Ext(Ring,Ideal) (missing documentation)
• Ext(Ring,Module) (missing documentation)
• Ext(Ring,Ring) (missing documentation)
• 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
• Ext^ZZ(Ring,Ideal) (missing documentation)
• Ext^ZZ(Ring,Matrix) (missing documentation)
• Ext^ZZ(Ring,Module) (missing documentation)
• Ext^ZZ(Ring,Ring) (missing documentation)
• Fano(ZZ,Ideal,Ring) -- Fano scheme
• flattenRing(Ring) -- see flattenRing -- write a ring as a (quotient of a) polynomial ring
• frac(Ring) -- see frac -- construct a fraction field
• genera(Ring)
• 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)
• GF(Ring) -- see GF -- make a finite field
• heft(Ring) -- see heft -- heft vector of ring or monoid
• hilbertFunction(List,Ring) -- see hilbertFunction -- the Hilbert function
• hilbertFunction(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 of homomorphisms
• Hom(Module,Ring) -- see Hom -- module of homomorphisms
• Hom(Ring,Ideal) -- see Hom -- module of homomorphisms
• Hom(Ring,Module) -- see Hom -- module of homomorphisms
• Hom(Ring,Ring) -- see Hom -- 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
• isStandardGradedPolynomialRing(Ring) -- see isStandardGradedPolynomialRing -- Checks whether a ring is a polynomial ring over a field with variables of degree 1
• isWeylAlgebra(Ring) (missing documentation)
• 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
• 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
• Ideal ** Ring -- see Module ** Ring -- tensor product
• Module ** Ring -- tensor product
• Ring ** Ideal -- see Module ** Ring -- tensor product
• Ring ** Module -- see Module ** Ring -- tensor product
• module(Ring) -- make or get a module
• monoid(Ring) -- make or retrieve a monoid
• MonoidElement _ Ring (missing documentation)
• multidegree(Ring) -- see multidegree -- multidegree
• mutableIdentity(Ring,ZZ) -- make a mutable identity matrix
• mutableMatrix(Ring,ZZ,ZZ) -- make a mutable matrix filled with zeroes
• nefCone(Ring) (missing documentation)
• nefGenerators(Ring) (missing documentation)
• numgens(Ring) -- number of generators of a polynomial ring
• options(Ring) -- see options(Monoid) -- get values used for optional arguments
• poincare(Ring) -- see poincare -- assemble degrees of a ring, module, or ideal into a polynomial
• precision(Ring) -- see precision
• primaryDecomposition(Ring) -- see primaryDecomposition(Module) -- irredundant primary decomposition of a module
• Proj(Ring) -- see Proj -- 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(ZZ,Ring) -- get a random homogeneous element from a graded ring
• random(ZZ,Ring) -- get a random homogeneous element from a graded ring
• 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 _ 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 Monoid -- 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) -- see Spec -- make an affine variety
• String _ Ring -- get a ring variable by name
• 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
• terms(Ring,RingElement) -- see terms -- provide a list of terms of a polynomial
• toField(Ring) -- declare that a ring is a field
• Tor_ZZ(Matrix,Ring) (missing documentation)
• Tor_ZZ(Ideal,Ring) -- see Tor_ZZ(Module,Module) -- compute a Tor module
• Tor_ZZ(Module,Ring) -- see Tor_ZZ(Module,Module) -- compute a Tor module
• Tor_ZZ(Ring,Ideal) (missing documentation)
• Tor_ZZ(Ring,Matrix) (missing documentation)
• Tor_ZZ(Ring,Module) (missing documentation)
• Tor_ZZ(Ring,Ring) (missing documentation)
• truncate(List,Ring) -- see truncate(List,Module) -- truncation of the graded ring, ideal or module at a specified degree or set of degrees
• truncate(ZZ,Ring) -- see truncate(List,Module) -- truncation of the graded ring, ideal or module at a specified degree or set of degrees
• use(Ring) -- install ring variables and ring operations
• variety(Ring) -- the variety previously associated to a given ring
• 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.