Ideal -- the class of all ideals

Types of ideal:

• MonomialIdeal -- the class of all monomial ideals handled by the engine

Functions and methods returning an ideal:

• Ideal * Ring -- see * -- a binary operator, usually used for multiplication
• Ring * Ideal -- see * -- a binary operator, usually used for multiplication
• RingElement * Ideal -- see * -- a binary operator, usually used for multiplication
• annihilator(Ideal) -- see annihilator -- the annihilator ideal
• annihilator(Module) -- see annihilator -- the annihilator ideal
• annihilator(RingElement) -- see annihilator -- the annihilator ideal
• content(RingElement) -- see content -- the content of a polynomial
• content(RingElement,RingElement) -- see content -- the content of a polynomial
• Fano(ZZ,Ideal) -- see Fano -- compute the ideal of a Fano scheme in the Grassmannian
• Fano(ZZ,Ideal,Ring) -- see Fano -- compute the ideal of a Fano scheme in the Grassmannian
• fittingIdeal -- Fitting ideal of a module
• graphIdeal(RingMap) -- the ideal of the graph of the regular map corresponding to a ring map
• Grassmannian -- compute the ideal of the Grassmannian of linear subspaces of a vector space
• homogenize(Ideal,RingElement) -- see homogenize -- homogenize with respect to a variable
• ideal -- make an ideal
• Ideal * Ideal -- product of ideals
• Ideal + Ideal -- sum of ideals
• Ideal : Ideal
• Ideal : Number (missing documentation)
• Ideal : RingElement
• Ideal ^ ZZ -- power of an ideal
• ideal(List) -- make an ideal
• ideal(Sequence) -- see ideal(List) -- make an ideal
• ideal(Matrix) -- make an ideal
• ideal(Module) -- convert a module to an ideal
• ideal(Number) -- see ideal(RingElement) -- make an ideal
• ideal(RingElement) -- make an ideal
• ideal(String) (missing documentation)
• intersect(Ideal) -- see intersect(Ideal,Ideal) -- compute an intersection of a sequence of ideals or modules
• intersect(Ideal,Ideal) -- compute an intersection of a sequence of ideals or modules
• kernel(RingMap) -- kernel of a ringmap
• leadTerm(Ideal) -- get the ideal of greatest terms
• lift(Ideal,type of RingElement) -- see lift -- lift to another ring
• minimalPresentation(Ideal) -- compute a minimal presentation of the quotient ring defined by an ideal
• prune(Ideal) -- see minimalPresentation(Ideal) -- compute a minimal presentation of the quotient ring defined by an ideal
• minors(ZZ,Matrix) -- ideal generated by minors
• Ideal ** Ring -- see Module ** Ring -- tensor product
• Ring ** Ideal -- see Module ** Ring -- tensor product
• Module : Module
• permanents(ZZ,Matrix) -- see permanents -- ideal generated by square permanents of a matrix
• pfaffians -- ideal generated by Pfaffians
• preimage(RingMap,Ideal) -- see preimage -- preimage of a map
• quotient(Ideal,Ideal)
• quotient(Ideal,Number) (missing documentation)
• quotient(Ideal,RingElement)
• quotient(Module,Module)
• RingMap Ideal -- see RingMap RingElement -- apply a ring map
• saturate(Ideal) -- see saturate -- saturation of ideal or submodule
• saturate(Ideal,Ideal) -- see saturate -- saturation of ideal or submodule
• saturate(Ideal,RingElement) -- see saturate -- saturation of ideal or submodule
• Schubert -- compute the Plücker ideal of a Schubert variety
• substitute(Ideal,List) -- see substitute -- substituting values for variables
• substitute(Ideal,Matrix) -- see substitute -- substituting values for variables
• substitute(Ideal,Ring) -- see substitute -- substituting values for variables
• substitute(Ideal,RingFamily) -- see substitute -- substituting values for variables
• trim(Ideal) -- see trim -- minimize generators and relations
• truncate(List,Ideal)
• truncate(List,Ring)

Methods that use an ideal:

• Number % Ideal -- see % -- a binary operator, usually used for remainder and reduction
• Ideal * Module -- see * -- a binary operator, usually used for multiplication
• Ideal * Vector -- see * -- a binary operator, usually used for multiplication
• Ideal + Number -- see + -- a unary or binary operator, usually used for addition
• Ideal + RingElement -- see + -- a unary or binary operator, usually used for addition
• Number + Ideal -- see + -- a unary or binary operator, usually used for addition
• RingElement + Ideal -- see + -- a unary or binary operator, usually used for addition
• basis(Ideal) -- see basis -- basis or generating set of all or part of a ring, ideal or module
• basis(InfiniteNumber,InfiniteNumber,Ideal) -- see basis -- basis or generating set of all or part of a ring, ideal or module
• basis(InfiniteNumber,List,Ideal) -- see basis -- basis or generating set of all or part of a ring, ideal or module
• basis(InfiniteNumber,ZZ,Ideal) -- see basis -- basis or generating set of all or part of a ring, ideal or module
• basis(List,Ideal) -- see basis -- basis or generating set of all or part of a ring, ideal or module
• basis(List,InfiniteNumber,Ideal) -- see basis -- basis or generating set of all or part of a ring, ideal or module
• basis(List,List,Ideal) -- see basis -- basis or generating set of all or part of a ring, ideal or module
• basis(List,ZZ,Ideal) -- see basis -- basis or generating set of all or part of a ring, ideal or module
• basis(ZZ,Ideal) -- see basis -- basis or generating set of all or part of a ring, ideal or module
• basis(ZZ,InfiniteNumber,Ideal) -- see basis -- basis or generating set of all or part of a ring, ideal or module
• basis(ZZ,List,Ideal) -- see basis -- basis or generating set of all or part of a ring, ideal or module
• basis(ZZ,ZZ,Ideal) -- see basis -- basis or generating set of all or part of a ring, ideal or module
• betti(Ideal) -- see betti -- display or modify a Betti diagram
• codim(Ideal) -- compute the codimension
• comodule(Ideal) -- see comodule -- submodule to quotient module
• quotient(Ideal) -- see comodule -- submodule to quotient module
• decompose(Ideal)
• degree(Ideal)
• degrees(Ideal) -- see degrees(Ring) -- degrees of generators
• dim(Ideal) -- compute the Krull dimension
• Ideal == Ideal -- see equality and containment
• Ideal == ZZ -- see equality and containment
• ZZ == Ideal -- see equality and containment
• euler(Ideal)
• eulers(Ideal)
• Ext(Ideal,Ideal) -- see Ext(Module,Module) -- total Ext module
• Ext(Ideal,Module) -- see Ext(Module,Module) -- total Ext module
• Ext(Ideal,Ring) -- see Ext(Module,Module) -- total Ext module
• Ext(Module,Ideal) -- see Ext(Module,Module) -- total Ext module
• Ext(Ring,Ideal) -- see Ext(Module,Module) -- total Ext module
• Ext^ZZ(Matrix,Ideal) -- see Ext^ZZ(Matrix,Module) -- map between Ext modules
• Ext^ZZ(Ideal,Matrix) -- see Ext^ZZ(Module,Matrix) -- map between Ext modules
• Ext^ZZ(Ideal,Ideal) -- see Ext^ZZ(Module,Module) -- Ext module
• Ext^ZZ(Ideal,Module) -- see Ext^ZZ(Module,Module) -- Ext module
• Ext^ZZ(Ideal,Ring) -- see Ext^ZZ(Module,Module) -- Ext module
• Ext^ZZ(Module,Ideal) -- see Ext^ZZ(Module,Module) -- Ext module
• Ext^ZZ(Ring,Ideal) -- see Ext^ZZ(Module,Module) -- Ext module
• flattenRing(Ideal) -- see flattenRing -- write a ring as a (quotient of a) polynomial ring
• gb(Ideal) -- see gb -- compute a Gröbner basis
• gbRemove(Ideal) -- see gbRemove -- remove Gröbner basis
• gbSnapshot(Ideal) -- see gbSnapshot -- the Gröbner basis matrix as so far computed
• genera(Ideal)
• generator(Ideal) -- see generator -- provide a single generator
• Ideal _ ZZ -- see generators of rings, ideals, and modules
• Ideal _* -- see generators of rings, ideals, and modules
• generators(Ideal) -- the generator matrix of an ideal
• genus(Ideal)
• groebnerBasis(Ideal) -- see groebnerBasis -- Gröbner basis, as a matrix
• hilbertFunction(Ideal) -- see hilbertFunction -- the Hilbert function
• hilbertFunction(List,Ideal) -- see hilbertFunction -- the Hilbert function
• hilbertFunction(ZZ,Ideal) -- see hilbertFunction -- the Hilbert function
• hilbertPolynomial(Ideal) -- compute the Hilbert polynomial of the quotient of the ambient ring by the ideal
• hilbertSeries(Ideal) -- compute the Hilbert series of the quotient of the ambient ring by the ideal
• Hom(Ideal,Ideal) -- see Hom -- module of homomorphisms
• Hom(Ideal,Module) -- see Hom -- module of homomorphisms
• Hom(Ideal,Ring) -- see Hom -- module of homomorphisms
• Hom(Module,Ideal) -- see Hom -- module of homomorphisms
• Hom(Ring,Ideal) -- see Hom -- module of homomorphisms
• Function \ Ideal -- see Ideal / Function -- apply a function to generators of an ideal
• Ideal / Function -- apply a function to generators of an ideal
• Ideal / Ideal -- quotient module
• Ideal ^ Array -- bracket power of an ideal
• Ideal ^~
• independentSets(Ideal) -- see independentSets -- some size-maximal independent subsets of variables modulo an ideal
• installHilbertFunction(Ideal,RingElement) (missing documentation)
• irreducibleCharacteristicSeries(Ideal) -- see irreducibleCharacteristicSeries -- irreducible characteristic series of an ideal
• isHomogeneous(Ideal) -- see isHomogeneous -- whether something is homogeneous (graded)
• isIdeal(Ideal) -- see isIdeal -- whether something is an ideal
• isMember(Number,Ideal) -- see isMember(RingElement,Ideal) -- test membership in an ideal
• isMember(RingElement,Ideal) -- test membership in an ideal
• isMonomialIdeal(Ideal) -- see isMonomialIdeal -- whether something is a monomial ideal
• isPrime(Ideal)
• isSquareFree(Ideal) -- see isSquareFree -- whether something is square free monomial ideal
• isSubset(Ideal,Ideal) -- whether one object is a subset of another
• isSubset(Ideal,Module) -- see isSubset(Module,Module) -- whether one object is a subset of another
• isSubset(Module,Ideal) -- see isSubset(Module,Module) -- whether one object is a subset of another
• jacobian(Ideal) -- the Jacobian matrix of the generators of an ideal
• leadTerm(ZZ,Ideal) -- get the ideal of lead polynomials
• lift(Ideal,type of QQ) -- see lift -- lift to another ring
• lift(Ideal,type of ZZ) -- see lift -- lift to another ring
• Matrix % Ideal -- see methods for normal forms and remainder -- normal form of ring elements and matrices
• RingElement % Ideal -- see methods for normal forms and remainder -- normal form of ring elements and matrices
• Vector % Ideal -- see methods for normal forms and remainder -- normal form of ring elements and matrices
• mingens(Ideal) -- see mingens(Module) -- minimal generator matrix
• minimalBetti(Ideal) -- see minimalBetti -- minimal betti numbers of (the minimal free resolution of) a homogeneous ideal or module
• Module / Ideal -- see Module / Module -- quotient module
• Module : Ideal
• Ideal _ List -- see Module _ List -- map from free module to some generators
• module(Ideal) (missing documentation)
• monomialIdeal(Ideal) -- monomial ideal of lead monomials of a Gröbner basis
• monomialSubideal(Ideal) -- see monomialSubideal -- find the largest monomial ideal in an ideal
• multidegree(Ideal) -- see multidegree -- multidegree
• normalCone(Ideal)
• normalCone(Ideal,RingElement)
• numgens(Ideal) -- number of generators of an ideal
• part(List,Ideal) (missing documentation)
• part(ZZ,Ideal) (missing documentation)
• poincare(Ideal) -- see poincare -- assemble degrees of a ring, module, or ideal into a polynomial
• quotient(Module,Ideal)
• random(List,Ideal) -- see random(ZZ,Ideal) -- get a random homogeneous element from a graded ideal
• random(ZZ,Ideal) -- get a random homogeneous element from a graded ideal
• randomKRationalPoint(Ideal) -- see randomKRationalPoint -- pick a random K rational point on the scheme X defined by I
• regularity(Ideal) -- see regularity -- compute the Castelnuovo-Mumford regularity
• ring(Ideal) -- see ring -- get the associated ring of an object
• Ring / Ideal -- make a quotient ring
• saturate(Module,Ideal) -- see saturate -- saturation of ideal or submodule
• saturate(Vector,Ideal) -- see saturate -- saturation of ideal or submodule
• singularLocus(Ideal) -- see singularLocus -- singular locus
• substitute(Ideal,Option) -- see substitute -- substituting values for variables
• support(Ideal) -- list of variables occurring in the generators of an ideal
• Tor_ZZ(Ideal,Ideal) -- see Tor_ZZ(Module,Module) -- compute a Tor module
• Tor_ZZ(Ideal,Module) -- see Tor_ZZ(Module,Module) -- compute a Tor module
• Tor_ZZ(Ideal,Ring) -- see Tor_ZZ(Module,Module) -- compute a Tor module
• Tor_ZZ(Module,Ideal) -- see Tor_ZZ(Module,Module) -- compute a Tor module
• Tor_ZZ(Ring,Ideal) -- see Tor_ZZ(Module,Module) -- compute a Tor module
• truncate(ZZ,Ideal)

For the programmer

The object Ideal is a type, with ancestor classes HashTable < Thing.

The source of this document is in Macaulay2Doc/doc_ideals.m2:110:0.