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
• ZZ % MonomialIdeal -- 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
• RingElement // RingElement -- see // -- a binary operator, usually used for quotient
• ZZ // MonomialIdeal -- see // -- a binary operator, usually used for quotient
• 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 rings, ideals, and modules
• 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
• homogenize(RingElement,RingElement,List) -- see homogenize -- homogenize with respect to a variable
• IndexedVariable _ Ring -- get a ring variable by name
• leadCoefficient(RingElement) -- see leadCoefficient -- the coefficient of the leading term
• leadMonomial(RingElement) -- see leadMonomial -- the leading monomial of a ring element
• leadTerm(RingElement) -- get the greatest term
• leadTerm(ZZ,RingElement) -- get the lead polynomials using part of the monomial order
• RingElement % GroebnerBasis -- see Matrix % GroebnerBasis -- compute the normal form of a ring element or matrix using a Gröbner basis
• Matrix _ Sequence -- get entry of matrix
• RingElement % MonomialIdeal -- see methods for normal forms and remainder -- normal form of ring elements and matrices
• 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)
• poincare -- assemble degrees of a ring, module, or ideal into a polynomial
• poincareN -- assemble degrees into polynomial
• RingElement _ Ring -- see promote -- promote to another ring
• promote(IndexedVariable,type of RingElement) (missing documentation)
• pseudoRemainder(RingElement,RingElement) -- see pseudoRemainder -- compute the pseudo-remainder
• RingElement // MonomialIdeal -- see quotient(Matrix,Matrix) -- factor a map through another with the same target
• 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
• RingElement / RingElement -- fraction
• RingElement ^ ZZ -- power
• RingMap RingElement -- apply a ring map
• 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 * 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
• + RingElement -- see + -- a unary or binary operator, usually used for addition
• Constant + RingElement -- see + -- a unary or binary operator, usually used for addition
• Ideal + RingElement -- see + -- a unary or binary operator, usually used for addition
• InexactNumber + RingElement -- see + -- a unary or binary operator, usually used for addition
• Matrix + RingElement -- see + -- a unary or binary operator, usually used for addition
• Number + RingElement -- see + -- a unary or binary operator, usually used for addition
• RingElement + Constant -- see + -- a unary or binary operator, usually used for addition
• RingElement + Ideal -- see + -- a unary or binary operator, usually used for addition
• RingElement + InexactNumber -- see + -- a unary or binary operator, usually used for addition
• RingElement + Matrix -- see + -- a unary or binary operator, usually used for addition
• RingElement + Number -- see + -- a unary or binary operator, usually used for addition
• RingElement + Vector -- see + -- a unary or binary operator, usually used for addition
• Vector + RingElement -- see + -- a unary or binary operator, usually used for addition
• Constant - RingElement -- see - -- a unary or binary operator, usually used for negation or subtraction
• InexactNumber - 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
• Number - RingElement -- see - -- a unary or binary operator, usually used for negation or subtraction
• RingElement - Constant -- see - -- a unary or binary operator, usually used for negation or subtraction
• RingElement - InexactNumber -- 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
• RingElement - Number -- see - -- a unary or binary operator, usually used for negation or subtraction
• RingElement - Vector -- see - -- a unary or binary operator, usually used for negation or subtraction
• Vector - RingElement -- see - -- a unary or binary operator, usually used for negation or subtraction
• List // RingElement -- see // -- a binary operator, usually used for quotient
• Number // RingElement -- see // -- a binary operator, usually used for quotient
• RingElement // Number -- see // -- a binary operator, usually used for quotient
• 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) -- see binomial -- binomial coefficient
• clean(RR,RingElement) -- see clean -- set to zero elements that are approximately zero
• coefficients(RingElement) -- see coefficients -- get the monomials and coefficients of a polynomial or matrix
• cokernel(RingElement) -- see cokernel(Matrix) -- cokernel of a map of modules
• columnAdd(MutableMatrix,ZZ,RingElement,ZZ) -- see columnAdd -- add a multiple of one column to another
• columnMult(MutableMatrix,ZZ,RingElement) -- see columnMult -- multiply a column by a ring element
• 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
• 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
• exp(RingElement)
• exponents(RingElement) -- see exponents -- the exponents of a polynomial
• factor(RingElement) -- factor a ring element
• fraction(RingElement,RingElement) -- see fraction
• gcd(RingElement) -- see gcd -- greatest common divisor
• gcd(RingElement,ZZ) -- see gcd -- greatest common divisor
• gcd(ZZ,RingElement) -- see gcd -- greatest common divisor
• gcdCoefficients(RingElement,RingElement) -- see gcdCoefficients -- greatest common divisor with coefficients
• gcdCoefficients(RingElement,ZZ) -- see gcdCoefficients -- greatest common divisor with coefficients
• gcdCoefficients(ZZ,RingElement) -- see gcdCoefficients -- greatest common divisor 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
• Ideal : RingElement
• ideal(RingElement) -- make an ideal
• image(RingElement) -- see image(Matrix) -- image of a map
• index(RingElement) -- see index -- numeric index of a ring variable
• indices(RingElement) -- indices of variables occurring in a polynomial
• installHilbertFunction(Ideal,RingElement) (missing documentation)
• installHilbertFunction(Matrix,RingElement) (missing documentation)
• installHilbertFunction(Module,RingElement) (missing documentation)
• isConstant(RingElement) -- see isConstant -- whether a ring element is constant
• isHomogeneous(RingElement) -- see isHomogeneous -- whether something is homogeneous (graded)
• isMember(RingElement,Ideal) -- test membership in an ideal
• isNormal(RingElement) (missing documentation)
• isPrime(RingElement) -- see isPrime -- whether a integer or polynomial is prime
• isUnit(RingElement) -- see isUnit -- whether a ring element is a unit
• jacobian(RingElement) -- see jacobian(Matrix) -- the matrix of partial derivatives of polynomials in a matrix
• kernel(RingElement) -- see kernel(Matrix) -- kernel of a matrix
• lcm(RingElement) -- see lcm -- least common multiple
• 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) -- see lift -- lift to another ring
• lift(MutableMatrix,type of RingElement) -- see lift -- lift to another ring
• lift(Vector,type of RingElement) -- see lift -- lift to another ring
• 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
• 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
• List / RingElement -- see Matrix / Number -- scalar division
• Matrix / RingElement -- see Matrix / Number -- scalar division
• Vector / RingElement -- see Matrix / Number -- scalar division
• 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
• Module / RingElement -- see Module / Module -- quotient module
• Module : RingElement
• MonomialIdeal : RingElement
• monomialIdeal(RingElement) -- see monomialIdeal(Matrix) -- monomial ideal of lead monomials
• monomials(RingElement) -- see monomials -- matrix of monomials in a ring element or matrix
• norm(InfiniteNumber,RingElement) -- see norm
• norm(RingElement) -- see norm
• norm(RR,RingElement) -- see norm
• normalCone(Ideal,RingElement)
• parts(RingElement) -- see parts -- display terms of a polynomial degree by degree
• precision(RingElement) -- see precision
• promote(Constant,type of RingElement) (missing documentation)
• quotient(Ideal,RingElement)
• Matrix // RingElement -- see quotient(Matrix,Matrix) -- factor a map through another with the same target
• RingElement // GroebnerBasis -- see quotient(Matrix,Matrix) -- factor a map through another with the same target
• RingElement // Matrix -- see quotient(Matrix,Matrix) -- factor a map through another with the same target
• quotient(Module,RingElement)
• quotient(MonomialIdeal,RingElement)
• 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
• ring(RingElement) -- see ring -- get the associated ring of an object
• Ring / RingElement -- see Ring / Ideal -- make a quotient ring
• RingElement .. RingElement -- a sequence of consecutive generators of a polynomial ring
• RingElement ..< RingElement -- a sequence of consecutive generators of a polynomial ring
• RingElement Number -- see RingElement Sequence -- evaluation of polynomials
• RingElement RingElement -- see RingElement Sequence -- evaluation of polynomials
• RingElement Sequence -- evaluation of polynomials
• roots(RingElement) -- compute the roots of a polynomial
• rowAdd(MutableMatrix,ZZ,RingElement,ZZ) -- see rowAdd -- add a multiple of one row to another
• 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
• size(RingElement) -- see size -- the size of an object
• 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
• 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
• topCoefficients(RingElement) -- see topCoefficients -- first variable and its coefficient of a polynomial or matrix
• vector(Module,RingElement) -- see vector -- make a vector
• vector(Ring,RingElement) -- see vector -- make a vector
• vector(RingElement) -- see vector -- make a vector
• vector(RingFamily,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.

The source of this document is in Macaulay2Doc/doc_rings.m2:177:0.