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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
• chi(CoherentSheaf)
• 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
• "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
• 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
• "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
• discriminant(RingElement,RingElement)
• "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
• 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 (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
• /// 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
• /// RingElement \\ Matrix /// -- see Matrix // Matrix -- factor a map through another
• "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 + 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 _ Monoid (missing documentation)
• RingElement _ Thing (missing documentation)
• RingElement Array -- substitution of variables
• "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
• "variety(RingElement)" -- see variety(Ring) -- the variety previously associated to a given ring
• "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.