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Ring -- the class of all rings

Description

Common ways to make a ring: 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:

See also

Menu

Types of ring:

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

Functions and methods returning a 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
  • 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
  • char(Ring) -- see char -- get the characteristic of the ring or field
  • coefficientRing(Ring) -- see coefficientRing -- get the coefficient ring
  • degree(Ring)
  • degreeGroup(Ring) -- see degreeGroup -- the degree group of a ring or monoid
  • 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
  • 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) -- see Ext(Module,Module) -- total Ext module
  • Ext(Ring,Module) -- see Ext(Module,Module) -- total Ext module
  • Ext(Ring,Ring) -- see Ext(Module,Module) -- total Ext module
  • Ext^ZZ(Matrix,Ring) -- see Ext^ZZ(Matrix,Module) -- map between Ext modules
  • Ext^ZZ(Ring,Matrix) -- see Ext^ZZ(Module,Matrix) -- 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) -- see Ext^ZZ(Module,Module) -- Ext module
  • Ext^ZZ(Ring,Module) -- see Ext^ZZ(Module,Module) -- Ext module
  • Ext^ZZ(Ring,Ring) -- see Ext^ZZ(Module,Module) -- Ext module
  • Fano(ZZ,Ideal,Ring) -- see Fano -- compute the ideal of a Fano scheme in the Grassmannian
  • flattenRing(Ring) -- see flattenRing -- write a ring as a (quotient of a) polynomial ring
  • frac(Ring) -- see frac -- construct a fraction field
  • genera(Ring)
  • Ring _* -- see generators of rings, ideals, and modules
  • 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)
  • Ring _ List -- see get a monomial by exponent vector -- make a monomial from a list of exponents
  • Ring _ ZZ -- see get a ring variable by index -- get a ring variable by index
  • String _ Ring -- see get a ring variable by name -- get a ring variable by name
  • 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
  • ideal(Ring) -- get 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
  • isFinitePrimeField(Ring) -- see isFinitePrimeField -- whether a ring is a finite prime field
  • isHomogeneous(Ring) -- see isHomogeneous -- whether something is homogeneous (graded)
  • isNormal(Ring)
  • 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
  • Number ^ Ring -- see lift -- lift to another 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
  • multidegree(Ring) -- see multidegree -- multidegree
  • mutableIdentity(Ring,ZZ) -- make a mutable identity matrix
  • mutableMatrix(Ring,List) -- see mutableMatrix -- make a mutable matrix
  • mutableMatrix(Ring,ZZ,ZZ) -- make a mutable matrix filled with zeroes
  • numgens(Ring) -- number of generators of a polynomial ring
  • options(Ring) -- see options(Monoid) -- get values used for optional arguments
  • part(List,Ring) (missing documentation)
  • part(ZZ,Ring) (missing documentation)
  • poincare(Ring) -- see poincare -- assemble degrees of a ring, module, or ideal into a polynomial
  • precision(Ring) -- see precision
  • 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 / 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 ^ BettiTally
  • Ring ^ List -- make a free module
  • Ring ^ ZZ -- make a free module
  • Ring ^~
  • Ring Array -- the standard way to make a polynomial ring
  • Ring List -- make a local polynomial ring
  • Ring Monoid -- make a polynomial ring
  • setupLift(Function,Ring,Ring) -- see setupLift -- set up lift from one ring to another
  • singularLocus(Ring) -- see singularLocus -- singular locus
  • 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(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) -- see Tor_ZZ(Module,Module) -- compute a Tor module
  • Tor_ZZ(Ring,Module) -- see Tor_ZZ(Module,Module) -- compute a Tor module
  • Tor_ZZ(Ring,Ring) -- see Tor_ZZ(Module,Module) -- compute a Tor module
  • truncate(List,Ring)
  • truncate(ZZ,Ring)
  • use(Ring) -- install ring variables and ring operations
  • vars(Ring) -- row matrix of the variables
  • vector(Ring,List) -- see vector -- make a vector
  • vector(Ring,Matrix) -- see vector -- make a vector
  • vector(Ring,Number) -- see vector -- make a vector
  • vector(Ring,RingElement) -- see vector -- make a vector
  • isWeylAlgebra(Ring) -- see Weyl algebras

Protected 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.


The source of this document is in Macaulay2Doc/doc_rings.m2:119:0.