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RingElement -- the class of all ring elements handled by the engine

Functions and methods returning a ring element :

• Number % GroebnerBasis -- see % -- a binary operator, usually used for remainder and reduction
• RingElement * RingElement -- see * -- a binary operator, usually used for multiplication
• RingElement + RingElement -- see + -- a unary or binary operator, usually used for addition
• - RingElement -- see - -- a unary or binary operator, usually used for negation or subtraction
• RingElement - RingElement -- see - -- a unary or binary operator, usually used for negation or subtraction
• coefficient -- coefficient of a monomial
• contract(Number,RingElement) -- see contract(Matrix,Matrix) -- contract a matrix by a matrix
• contract(RingElement,Number) -- see contract(Matrix,Matrix) -- contract a matrix by a matrix
• contract(RingElement,RingElement) -- see contract(Matrix,Matrix) -- contract a matrix by a matrix
• determinant(Matrix) -- see determinant -- determinant of a matrix
• diff(RingElement,RingElement) -- differentiation
• discriminant(RingElement,RingElement)
• gcd(RingElement,RingElement) -- see gcd -- greatest common divisor
• generator(Ideal) -- see generator -- provide a single generator
• generator(Module) -- see generator -- provide a single generator
• Ideal _ ZZ -- see generators of ideals and modules
• homogenize(RingElement,RingElement,List) -- see homogenize -- homogenize with respect to a variable
• IndexedVariable _ Ring -- get a ring variable by name
• leadTerm(RingElement) -- get the greatest term
• leadTerm(ZZ,RingElement) -- get the lead polynomials using part of the monomial order
• RingElement % GroebnerBasis -- see Matrix % GroebnerBasis -- calculate the normal form of ring elements and matrices using a (partially computed) Gröbner basis
• RingElement // RingElement -- see Matrix // Matrix -- factor a map through another
• Matrix _ Sequence -- get entry of matrix
• RingElement % RingElement -- see methods for normal forms and remainder -- normal form of ring elements and matrices
• MonoidElement _ Ring (missing documentation)
• part(InfiniteNumber,InfiniteNumber,RingElement) -- see part -- select terms of a polynomial by degree(s) or weight(s)
• part(InfiniteNumber,InfiniteNumber,VisibleList,RingElement) -- see part -- select terms of a polynomial by degree(s) or weight(s)
• part(InfiniteNumber,ZZ,RingElement) -- see part -- select terms of a polynomial by degree(s) or weight(s)
• part(InfiniteNumber,ZZ,VisibleList,RingElement) -- see part -- select terms of a polynomial by degree(s) or weight(s)
• part(List,RingElement) -- see part -- select terms of a polynomial by degree(s) or weight(s)
• part(Nothing,Nothing,RingElement) -- see part -- select terms of a polynomial by degree(s) or weight(s)
• part(Nothing,Nothing,VisibleList,RingElement) -- see part -- select terms of a polynomial by degree(s) or weight(s)
• part(Nothing,ZZ,RingElement) -- see part -- select terms of a polynomial by degree(s) or weight(s)
• part(Nothing,ZZ,VisibleList,RingElement) -- see part -- select terms of a polynomial by degree(s) or weight(s)
• part(ZZ,InfiniteNumber,RingElement) -- see part -- select terms of a polynomial by degree(s) or weight(s)
• part(ZZ,InfiniteNumber,VisibleList,RingElement) -- see part -- select terms of a polynomial by degree(s) or weight(s)
• part(ZZ,Nothing,RingElement) -- see part -- select terms of a polynomial by degree(s) or weight(s)
• part(ZZ,Nothing,VisibleList,RingElement) -- see part -- select terms of a polynomial by degree(s) or weight(s)
• part(ZZ,RingElement) -- see part -- select terms of a polynomial by degree(s) or weight(s)
• part(ZZ,VisibleList,RingElement) -- see part -- select terms of a polynomial by degree(s) or weight(s)
• part(ZZ,ZZ,RingElement) -- see part -- select terms of a polynomial by degree(s) or weight(s)
• part(ZZ,ZZ,VisibleList,RingElement) -- see part -- select terms of a polynomial by degree(s) or weight(s)
• poincare -- assemble degrees of a ring, module, or ideal into a polynomial
• poincareN -- assemble degrees into polynomial
• poly(String) -- make a polynomial using classic Macaulay syntax
• RingElement _ Ring -- see promote -- promote to another ring
• pseudoRemainder(RingElement,RingElement) -- see pseudoRemainder -- compute the pseudo-remainder
• random(ZZ,Ideal) -- get a random homogeneous element from a graded ideal
• 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
• resultant -- see resultant(RingElement,RingElement,RingElement)
• Ring _ List -- make a monomial from a list of exponents
• Ring _ ZZ -- get a ring variable by index
• RingElement / RingElement -- fraction
• RingElement ^ ZZ -- power
• RingMap RingElement -- apply a ring map
• someTerms(RingElement,ZZ,ZZ) -- see someTerms -- select some terms of a polynomial
• String _ Ring -- get a ring variable by name
• substitute(Number,Ring) -- see substitute -- substituting values for variables
• substitute(Number,RingFamily) -- see substitute -- substituting values for variables
• substitute(RingElement,List) -- see substitute -- substituting values for variables
• substitute(RingElement,Matrix) -- see substitute -- substituting values for variables
• substitute(RingElement,Ring) -- see substitute -- substituting values for variables
• substitute(RingElement,RingFamily) -- see substitute -- substituting values for variables
• Symbol _ Ring -- get a ring variable by name
• trace(Matrix) -- trace of a matrix

Methods that use a ring element :

• Number % RingElement -- see % -- a binary operator, usually used for remainder and reduction
• RingElement % Number -- see % -- a binary operator, usually used for remainder and reduction
• Matrix * RingElement -- see * -- a binary operator, usually used for multiplication
• Ring * RingElement -- see * -- a binary operator, usually used for multiplication
• RingElement * ChainComplexMap -- see * -- a binary operator, usually used for multiplication
• RingElement * GradedModuleMap -- see * -- a binary operator, usually used for multiplication
• RingElement * Ideal -- see * -- a binary operator, usually used for multiplication
• RingElement * Matrix -- see * -- a binary operator, usually used for multiplication
• RingElement * Module -- see * -- a binary operator, usually used for multiplication
• RingElement * MonomialIdeal -- see * -- a binary operator, usually used for multiplication
• RingElement * MutableMatrix -- see * -- a binary operator, usually used for multiplication
• RingElement * Vector -- see * -- a binary operator, usually used for multiplication
• Vector * RingElement -- see * -- a binary operator, usually used for multiplication
• ChainComplexMap + RingElement -- see + -- a unary or binary operator, usually used for addition
• GradedModuleMap + RingElement -- see + -- a unary or binary operator, usually used for addition
• Ideal + RingElement -- see + -- a unary or binary operator, usually used for addition
• Matrix + RingElement -- see + -- a unary or binary operator, usually used for addition
• RingElement + ChainComplexMap -- see + -- a unary or binary operator, usually used for addition
• RingElement + GradedModuleMap -- see + -- a unary or binary operator, usually used for addition
• RingElement + Matrix -- see + -- a unary or binary operator, usually used for addition
• + RingElement (missing documentation)
• ChainComplexMap - RingElement -- see - -- a unary or binary operator, usually used for negation or subtraction
• GradedModuleMap - RingElement -- see - -- a unary or binary operator, usually used for negation or subtraction
• Matrix - RingElement -- see - -- a unary or binary operator, usually used for negation or subtraction
• RingElement - ChainComplexMap -- see - -- a unary or binary operator, usually used for negation or subtraction
• RingElement - GradedModuleMap -- see - -- a unary or binary operator, usually used for negation or subtraction
• RingElement - Matrix -- see - -- a unary or binary operator, usually used for negation or subtraction
• Number // RingElement -- see // -- a binary operator, usually used for quotient
• RingElement // Number -- see // -- a binary operator, usually used for quotient
• ChainComplexMap == RingElement -- see == -- equality
• GradedModuleMap == RingElement -- see == -- equality
• Matrix == RingElement -- see == -- equality
• Number == RingElement -- see == -- equality
• RingElement == ChainComplexMap -- see == -- equality
• RingElement == GradedModuleMap -- see == -- equality
• RingElement == Matrix -- see == -- equality
• RingElement == Number -- see == -- equality
• RingElement == RingElement -- see == -- equality
• RingElement == ZZ -- see == -- equality
• ZZ == RingElement -- see == -- equality
• annihilator(RingElement) -- see annihilator -- the annihilator ideal
• antipode(RingElement) -- see antipode -- antipode for skew commuting polynomial rings
• asin(RingElement) (missing documentation)
• atan(RingElement) (missing documentation)
• baseName(RingElement) -- see baseName -- the base name of a generator
• binomial(RingElement,ZZ) -- see binomial -- binomial coefficient
• clean(RR,RingElement) -- see clean -- Set to zero elements that are approximately zero
• coefficients(RingElement) -- see coefficients -- monomials and their coefficients
• cokernel(RingElement) -- see cokernel -- cokernel of a map of modules, graded modules, or chaincomplexes
• columnMult(MutableMatrix,ZZ,RingElement) -- see columnMult -- multiply a column by a ring element
• Constant * RingElement -- see Constant
• Constant + RingElement -- see Constant
• Constant - RingElement -- see Constant
• Constant / RingElement -- see Constant
• RingElement * Constant -- see Constant
• RingElement + Constant -- see Constant
• RingElement - Constant -- see Constant
• RingElement / Constant -- see Constant
• content(RingElement) -- see content -- the content of a polynomial
• content(RingElement,RingElement) -- see content -- the content of a polynomial
• contract(Matrix,RingElement) -- see contract(Matrix,Matrix) -- contract a matrix by a matrix
• contract(RingElement,Matrix) -- see contract(Matrix,Matrix) -- contract a matrix by a matrix
• contract(RingElement,Vector) -- see contract(Matrix,Matrix) -- contract a matrix by a matrix
• contract(Vector,RingElement) -- see contract(Matrix,Matrix) -- contract a matrix by a matrix
• cos(RingElement) (missing documentation)
• cosh(RingElement) (missing documentation)
• degree(RingElement)
• degree(RingElement,RingElement) -- degree with respect to a variable
• diff(RingElement,Vector) -- see diff(Matrix,Matrix) -- differentiate a matrix by a matrix
• diff(Vector,RingElement) -- see diff(Matrix,Matrix) -- differentiate a matrix by a matrix
• diff(Matrix,RingElement) -- differentiation
• diff(RingElement,Matrix) -- differentiate each entry of a matrix
• divideByVariable(Matrix,RingElement) -- see divideByVariable -- divide all columns by a (power of a) variable
• divideByVariable(Matrix,RingElement,ZZ) -- see divideByVariable -- divide all columns by a (power of a) variable
• dual(MonomialIdeal,RingElement) -- the Alexander dual
• eliminate(RingElement,Ideal) -- see eliminate
• exp(RingElement)
• expm1(RingElement) (missing documentation)
• exponents(RingElement) -- see exponents -- the exponents of a polynomial
• factor(RingElement) -- factor a ring element
• fraction(RingElement,RingElement) -- see fraction
• fromDividedPowers(RingElement) -- see fromDividedPowers -- Translates from divided power monomial basis to ordinary monomial basis
• fromDual(RingElement) -- see fromDual -- Ideal from inverse system
• gcd(RingElement,ZZ) -- see gcd -- greatest common divisor
• gcd(ZZ,RingElement) -- see gcd -- greatest common divisor
• gcdCoefficients(RingElement,RingElement) -- see gcdCoefficients -- gcd with coefficients
• genericMatrix(Ring,RingElement,ZZ,ZZ) -- see genericMatrix -- make a generic matrix of variables
• genericSkewMatrix(Ring,RingElement,ZZ) -- see genericSkewMatrix -- make a generic skew symmetric matrix of variables
• genericSymmetricMatrix(Ring,RingElement,ZZ) -- see genericSymmetricMatrix -- make a generic symmetric matrix
• homogenize(Ideal,RingElement) -- see homogenize -- homogenize with respect to a variable
• homogenize(Matrix,RingElement) -- see homogenize -- homogenize with respect to a variable
• homogenize(Matrix,RingElement,List) -- see homogenize -- homogenize with respect to a variable
• homogenize(Module,RingElement) -- see homogenize -- homogenize with respect to a variable
• homogenize(Module,RingElement,List) -- see homogenize -- homogenize with respect to a variable
• homogenize(RingElement,RingElement) -- see homogenize -- homogenize with respect to a variable
• homogenize(Vector,RingElement) -- see homogenize -- homogenize with respect to a variable
• homogenize(Vector,RingElement,List) -- see homogenize -- homogenize with respect to a variable
• icPIdeal(RingElement,RingElement,ZZ) -- see icPIdeal -- compute the integral closure in prime characteristic of a principal ideal
• Ideal * RingElement (missing documentation)
• ideal(RingElement) -- make an ideal
• idealizer(Ideal,RingElement) -- see idealizer -- compute Hom(I,I) as a quotient ring
• image(RingElement) -- see image -- image of a map
• index(RingElement) -- see index -- numeric index of a ring variable
• indices(RingElement) -- indices of variables occurring in a polynomial
• InexactNumber % RingElement (missing documentation)
• InexactNumber * RingElement (missing documentation)
• InexactNumber + RingElement (missing documentation)
• InexactNumber - RingElement (missing documentation)
• InexactNumber / RingElement (missing documentation)
• InexactNumber // RingElement (missing documentation)
• InexactNumber == RingElement (missing documentation)
• installHilbertFunction(Ideal,RingElement) (missing documentation)
• installHilbertFunction(Matrix,RingElement) (missing documentation)
• installHilbertFunction(Module,RingElement) (missing documentation)
• integralClosure(Ideal,RingElement) -- see integralClosure(Ideal,RingElement,ZZ) -- integral closure of an ideal in an affine domain
• integralClosure(Ideal,RingElement,ZZ) -- integral closure of an ideal in an affine domain
• inverseSystem(RingElement) -- see inverseSystem -- Inverse systems with equivariance
• isConstant(RingElement) -- see isConstant -- whether a ring element is constant
• isHomogeneous(RingElement) -- see isHomogeneous -- whether something is homogeneous (graded)
• isLinearType(Ideal,RingElement) -- see isLinearType -- Determine whether module has linear type
• isLinearType(Module,RingElement) -- see isLinearType -- Determine whether module has linear type
• isMember(RingElement,Ideal) -- test membership in an ideal
• isNormal(RingElement) (missing documentation)
• isPrime(RingElement) (missing documentation)
• isReduction(Ideal,Ideal,RingElement) -- see isReduction -- Determine whether an ideal is a reduction
• isReduction(Module,Module,RingElement) -- see isReduction -- Determine whether an ideal is a reduction
• isUnit(RingElement) -- see isUnit -- whether a ring element is a unit
• jacobian(RingElement) (missing documentation)
• kernel(RingElement) -- see kernel(Matrix) -- kernel of a matrix
• lcm(RingElement,RingElement) -- see lcm -- least common multiple
• lcm(RingElement,ZZ) -- see lcm -- least common multiple
• lcm(ZZ,RingElement) -- see lcm -- least common multiple
• lift(Ideal,type of RingElement) -- see lift -- lift to another ring
• lift(Matrix,type of RingElement) -- see lift -- lift to another ring
• lift(Module,type of RingElement) (missing documentation)
• lift(MutableMatrix,type of RingElement) (missing documentation)
• lift(RingElement,type of MonoidElement) (missing documentation)
• lift(Vector,type of RingElement) (missing documentation)
• liftable(Constant,type of RingElement) (missing documentation)
• List % RingElement (missing documentation)
• List / RingElement -- vector division
• List // RingElement (missing documentation)
• listForm(RingElement) -- see listForm -- convert to list form
• log1p(RingElement) (missing documentation)
• map(Module,Module,RingElement) -- construct the map induced by multiplication by a ring element on the generators
• Matrix ** RingElement -- a binary operator, usually used for tensor product or Cartesian product
• Matrix ++ RingElement -- see Matrix ++ Matrix -- direct sum of maps
• RingElement ++ Matrix -- see Matrix ++ Matrix -- direct sum of maps
• RingElement ++ RingElement -- see Matrix ++ Matrix -- direct sum of maps
• Matrix // RingElement -- see Matrix // Matrix -- factor a map through another
• RingElement // GroebnerBasis -- see Matrix // Matrix -- factor a map through another
• RingElement // Matrix -- see Matrix // Matrix -- factor a map through another
• RingElement // MonomialIdeal -- see Matrix // Matrix -- factor a map through another
• Matrix \\ RingElement (missing documentation)
• Matrix | RingElement -- see Matrix | Matrix -- join matrices horizontally
• RingElement | Matrix -- see Matrix | Matrix -- join matrices horizontally
• RingElement | RingElement -- see Matrix | Matrix -- join matrices horizontally
• Matrix || RingElement -- see Matrix || Matrix -- join matrices vertically
• RingElement || Matrix -- see Matrix || Matrix -- join matrices vertically
• RingElement || RingElement -- see Matrix || Matrix -- join matrices vertically
• matrix(RingElement) -- make a matrix from a ring element
• Matrix % RingElement -- 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
• RingElement % Matrix -- see methods for normal forms and remainder -- normal form of ring elements and matrices
• RingElement % MonomialIdeal -- see methods for normal forms and remainder -- normal form of ring elements and matrices
• Module * RingElement (missing documentation)
• Module / RingElement -- see Module / Module -- quotient module
• MonomialIdeal : RingElement (missing documentation)
• monomialIdeal(RingElement) -- see monomialIdeal(Matrix) -- monomial ideal of lead monomials
• monomials(RingElement) -- see monomials -- matrix of monomials in a ring element or matrix
• multiplicity(Ideal,RingElement) -- see multiplicity -- Compute the Hilbert-Samuel multiplicity of an ideal
• MutableMatrix * RingElement (missing documentation)
• norm(InfiniteNumber,RingElement) -- see norm
• norm(RingElement) -- see norm
• norm(RR,RingElement) -- see norm
• normalCone(Ideal,RingElement) -- see normalCone -- The normal cone of a subscheme
• parts(RingElement) -- see parts -- display terms of a polynomial degree by degree
• precision(RingElement) -- see precision
• Ideal : RingElement -- see quotient(Module,Module) -- ideal or submodule quotient
• Module : RingElement -- see quotient(Module,Module) -- ideal or submodule quotient
• quotient(Ideal,RingElement) -- see quotient(Module,Module) -- ideal or submodule quotient
• quotient(Module,RingElement) -- see quotient(Module,Module) -- ideal or submodule quotient
• quotient(MonomialIdeal,RingElement) -- see quotient(Module,Module) -- ideal or submodule quotient
• quotientRemainder(InexactNumber,RingElement) -- see quotientRemainder(RingElement,RingElement) -- quotient and remainder
• quotientRemainder(Number,RingElement) -- see quotientRemainder(RingElement,RingElement) -- quotient and remainder
• quotientRemainder(RingElement,InexactNumber) -- see quotientRemainder(RingElement,RingElement) -- quotient and remainder
• quotientRemainder(RingElement,Number) -- see quotientRemainder(RingElement,RingElement) -- quotient and remainder
• quotientRemainder(RingElement,RingElement) -- quotient and remainder
• radicalContainment(RingElement,Ideal) -- see radicalContainment -- whether an element is contained in the radical of an ideal
• reesAlgebra(Ideal,RingElement) -- see reesAlgebra -- Compute the defining ideal of the Rees Algebra
• reesAlgebra(Module,RingElement) -- see reesAlgebra -- Compute the defining ideal of the Rees Algebra
• reesIdeal(Ideal,RingElement) -- see reesIdeal -- Compute the defining ideal of the Rees Algebra
• reesIdeal(Module,RingElement) -- see reesIdeal -- Compute the defining ideal of the Rees Algebra
• resultant(RingElement,RingElement,RingElement)
• ring(RingElement) -- see ring -- get the associated ring of an object
• Ring / RingElement -- see Ring / Ideal -- make a quotient ring
• RingElement % InexactNumber (missing documentation)
• RingElement * InexactNumber (missing documentation)
• RingElement * Tally (missing documentation)
• RingElement * VirtualTally (missing documentation)
• RingElement + Ideal (missing documentation)
• RingElement + InexactNumber (missing documentation)
• RingElement - InexactNumber (missing documentation)
• RingElement .. RingElement -- a sequence of consecutive generators of a polynomial ring
• RingElement .. Thing (missing documentation)
• RingElement ..< RingElement -- a sequence of consecutive generators of a polynomial ring
• RingElement ..< Thing (missing documentation)
• RingElement / InexactNumber (missing documentation)
• RingElement // InexactNumber (missing documentation)
• RingElement == InexactNumber (missing documentation)
• RingElement \\ Matrix (missing documentation)
• RingElement _ Monoid (missing documentation)
• RingElement _ Thing (missing documentation)
• RingElement Number -- see RingElement Sequence -- evaluation of polynomials
• RingElement RingElement -- see RingElement Sequence -- evaluation of polynomials
• RingElement Sequence -- evaluation of polynomials
• ringFromFractions(Matrix,RingElement) -- see ringFromFractions -- find presentation for f.g. ring
• roots(RingElement) -- compute the roots of a polynomial
• rowMult(MutableMatrix,ZZ,RingElement) -- see rowMult -- multiply a row by a ring element
• saturate(Ideal,RingElement) -- see saturate -- saturation of ideal or submodule
• saturate(Module,RingElement) -- see saturate -- saturation of ideal or submodule
• saturate(MonomialIdeal,RingElement) -- see saturate -- saturation of ideal or submodule
• saturate(Vector,RingElement) -- see saturate -- saturation of ideal or submodule
• sec(RingElement) (missing documentation)
• sech(RingElement) (missing documentation)
• sin(RingElement) (missing documentation)
• sinh(RingElement) (missing documentation)
• size(RingElement) -- see size -- the size of an object
• specialFiber(Ideal,RingElement) -- see specialFiber -- Special fiber of a blowup
• specialFiber(Module,RingElement) -- see specialFiber -- Special fiber of a blowup
• specialFiberIdeal(Ideal,RingElement) -- see specialFiberIdeal -- Special fiber of a blowup
• specialFiberIdeal(Module,RingElement) -- see specialFiberIdeal -- Special fiber of a blowup
• standardForm(RingElement) -- see standardForm -- convert to standard form
• substitute(RingElement,Option) -- see substitute -- substituting values for variables
• support(RingElement) -- see support -- list of variables occurring in a polynomial or matrix
• sylvesterMatrix(RingElement,RingElement,RingElement)
• tan(RingElement) (missing documentation)
• tanh(RingElement) (missing documentation)
• terms(Ring,RingElement) -- see terms -- provide a list of terms of a polynomial
• terms(RingElement) -- see terms -- provide a list of terms of a polynomial
• testHunekeQuestion(RingElement) -- see testHunekeQuestion -- tests a conjecture on integral closures strengthening the Eisenbud-Mazur conjecture
• Thing .. RingElement (missing documentation)
• Thing ..< RingElement (missing documentation)
• toDividedPowers(RingElement) -- see toDividedPowers -- Translates to divided power monomial basis from ordinary monomial basis
• topCoefficients(RingElement) -- see topCoefficients -- first variable and its coefficient of a polynomial or matrix
• vector(Module,RingElement) -- see vector -- make a vector
• vector(RingElement) -- see vector -- make a vector
• weightRange(List,RingElement) -- see weightRange -- the pair of lowest and highest weights of the monomials
• weightRange(RingElement) -- see weightRange -- the pair of lowest and highest weights of the monomials

For the programmer

The object RingElement is a type, with ancestor classes BasicList < Thing.