Ring -- the class of all 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
• coefficientRing(Ring), see coefficientRing -- get the coefficient ring
• integralClosure(Ring) -- compute the integral closure of a ring
• modifyRing(Ring), see modifyRing -- make a copy of a ring, with some features changed
• ring -- Get the associated ring of an object
• Ring ** Ring -- tensor product
• coefficientRing(SchurRing), see SchurRing -- the class of all Schur rings
• symmetricAlgebra(Module) -- the symmetric algebra of a module
• tensor(Ring,Ring), see tensor -- 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
• AffineVariety ** Ring, see ** -- a binary operator, usually used for tensor product or Cartesian product
• ChainComplex ** Ring, see ** -- a binary operator, usually used for tensor product or Cartesian product
• ProjectiveVariety ** Ring, see ** -- a binary operator, usually used for tensor product or Cartesian product
• allGenerators(Ring), see allGenerators -- list of all generators
• 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) -- make an empty chain complex over a ring
• char(Ring), see char -- computes the characteristic of a field or ring
• degree(Ring)
• degreeLength(Ring), see degreeLength -- the number of degrees
• degrees(Ring) -- degrees of generators
• degreesRing(Ring) -- the ring of degrees
• diagonalMatrix(Ring,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
• 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) -- make a finite field from a ring
• hilbertFunction(List,Ring) -- compute the Hilbert function of a ring
• hilbertFunction(ZZ,Ring), see hilbertFunction(List,Ring) -- compute the Hilbert function of a ring
• hilbertPolynomial(Ring) -- compute the Hilbert polynomial of the ring
• hilbertSeries(Ring) -- compute the Hilbert series 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
• ICfractions(Ring), see ICfractions -- Compute the fractions integral over a 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) -- determine whether 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
• map(Ring,Matrix) -- make a ring map
• map(Ring,Ring) -- map which associates variables, rest maps to zero
• map(Ring,Ring,List) -- make a ring map
• map(Ring,Ring,Matrix) -- make a ring map
• Matrix ** Ring -- tensor product
• matrix(Ring,List) -- create a matrix from a doubly nested list of ring elements or matrices
• minimalPresentation(Ring) -- compute a minimal presentation of a quotient ring
• Module ** Ring -- tensor product
• module(Ring)
• mutableIdentity(Ring,ZZ) -- make a mutable identity matrix
• mutableZero(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
• Proj(Ring) -- make a projective variety
• random(List,Ring) -- a random ring element of a given degree
• random(Ring) -- random element of a ring
• random(ZZ,Ring) -- a random ring element of a given degree
• Ring ** Matrix -- missing documentation
• Ring ** Module -- missing documentation
• 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 _ IndexedVariable -- missing documentation
• Ring _ List -- make a monomial from a list of exponents
• Ring _ String -- get a ring variable by name
• Ring _ Symbol -- missing documentation
• Ring _ ZZ -- get a ring variable by index
• Ring ~ -- make the structure sheaf
• Ring Array -- the standard way to make a polynomial ring
• Ring OrderedMonoid -- make a polynomial ring
• sheaf(Ring) -- make the structure sheaf
• 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(RingElement,Ring), see substitute -- substituting values for variables
• substitute(Vector,Ring), see substitute -- substituting values for variables
• substitute(Number,Ring) -- missing documentation
• RingElement _ Ring, see Symbol _ Ring -- get a ring variable by name
• Symbol _ Ring -- get a ring variable by name
• tensor(Ring,RingMap,Matrix) -- missing documentation
• tensor(Ring,RingMap,Module) -- missing documentation
• use(Ring) -- install ring variables and ring operations
• vars(Ring) -- row matrix of the variables
• ZZ _ Ring -- missing documentation

Fixed objects of class Ring :

• CC -- the class of all complex numbers
• CCC -- high-precision complex numbers
• QQ -- the class of all rational numbers
• RR -- the class of all real numbers
• RRR -- high-precision real numbers
• ZZ -- the class of all integers

For the programmer

The object Ring is a type, with ancestor classes Type < MutableHashTable < HashTable < Thing.