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

## Functions and methods returning a ring element :

• 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) (missing documentation) -- compute the Euler characteristic of a coherent sheaf
• 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
• exp(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
• 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)
• poly(String) -- make a polynomial using classic Macaulay syntax
• pseudoRemainder(RingElement,RingElement), see pseudoRemainder -- compute the pseudo-remainder
• random(List,Ring), see random(Type) -- random element of a type
• random(ZZ,Ring), see random(Type) -- random element of a type
• Ring _ List -- make a monomial from a list of exponents
• Ring _ String -- get a ring variable by name
• Ring _ ZZ -- get a ring variable by index
• RingElement / RingElement -- fraction
• RingElement ^ ZZ -- power
• RingMap RingElement -- apply a ring map
• round(ZZ,RingElement) (missing documentation)
• someTerms(RingElement,ZZ,ZZ), see someTerms -- select some terms of a polynomial
• 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
• baseName(RingElement), see baseName -- the base name of a generator
• binomial(RingElement,ZZ) (missing documentation)
• 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) -- 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
• 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
• exponents(RingElement), see exponents -- list the exponents in 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), see installHilbertFunction -- install a Hilbert function without computation
• installHilbertFunction(Matrix,RingElement), see installHilbertFunction -- install a Hilbert function without computation
• installHilbertFunction(Module,RingElement), see installHilbertFunction -- install a Hilbert function without computation
• 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
• 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
• 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
• RingElement ^ Ring, see lift -- lift to another ring
• RingElement ^ RingFamily, see lift -- lift to another ring
• lift(Module,type of RingElement) (missing documentation)
• lift(MutableMatrix,type of RingElement) (missing documentation)
• List % RingElement (missing documentation)
• List // RingElement (missing documentation)
• listForm(RingElement), see listForm -- convert to list form
• 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
• member(RingElement,Ideal) (missing documentation)
• 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
• projectiveHilbertPolynomial(RingElement) (missing documentation)
• RingElement _ Ring, see promote -- promote to another ring
• promote(Module,type of RingElement) (missing documentation)
• promote(MutableMatrix,type of RingElement) (missing documentation)
• promote(Vector,type of RingElement) (missing documentation)
• pseudoRemainder(RingElement,RingElement,RingElement) (missing documentation) -- pseudo-remainder
• Ideal : RingElement, see quotient(Ideal,Ideal) -- ideal or submodule quotient
• Module : RingElement, see quotient(Ideal,Ideal) -- ideal or submodule quotient
• quotient(Ideal,RingElement), see quotient(Ideal,Ideal) -- ideal or submodule quotient
• quotient(Module,RingElement), see quotient(Ideal,Ideal) -- ideal or submodule quotient
• quotient(MonomialIdeal,RingElement) (missing documentation)
• quotientRemainder(InexactNumber,RingElement) (missing documentation)
• quotientRemainder(RingElement,InexactNumber) (missing documentation)
• quotientRemainder(Number,RingElement), see quotientRemainder(RingElement,RingElement) -- quotient and remainder
• quotientRemainder(RingElement,Number), see quotientRemainder(RingElement,RingElement) -- quotient and remainder
• quotientRemainder(RingElement,RingElement) -- quotient and remainder
• 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 ~ (missing documentation)
• RingElement Array -- substitution of variables
• RingElement RingElement (missing documentation) -- Multiplication in the Ext-algebra
• 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) (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
• switch(RingElement) (missing documentation)
• sylvesterMatrix(RingElement,RingElement,RingElement)
• 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
• 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
• width(RingElement) (missing documentation)

## For the programmer

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