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
• 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), see homogenize -- homogenize with respect to a variable
• 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 leading coefficient
• leadMonomial(RingElement), see leadMonomial -- the leading monomial
• 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) Groebner basis
• RingElement // RingElement, see Matrix // Matrix -- factor a map through another
• Matrix _ Sequence -- missing documentation
• RingElement % RingElement, see methods for normal forms and remainder -- calculate the normal form of ring elements and matrices
• part(List,RingElement), see part -- sum of monomials of a polynomial of a given degree
• part(ZZ,RingElement), see part -- sum of monomials of a polynomial of a given degree
• poly(String) -- make a polynomial using classic Macaulay syntax
• pseudoRemainder(RingElement,RingElement), see pseudoRemainder -- compute the pseudo-remainder
• random(List,Ring) -- a random ring element of a given degree
• random(Ring) -- random element of a ring
• random(ZZ,Ring) -- a random ring element of a given degree
• Ring _ List -- make a monomial from a list of exponents
• Ring _ String -- get a ring variable by name
• Ring _ Symbol -- missing documentation
• 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
• 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(Number,Ring) -- missing documentation
• RingElement _ Ring, see Symbol _ Ring -- get a ring variable by name
• Symbol _ Ring -- get a ring variable by name
• trace(Matrix) -- trace of a matrix
• ZZ _ Ring -- missing documentation

Methods that use a ring element :

• 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
• Matrix ** RingElement, see ** -- a binary operator, usually used for tensor product or Cartesian product
• RingElement ** Matrix, see ** -- a binary operator, usually used for tensor product or Cartesian product
• RingElement ** RingElement, see ** -- a binary operator, usually used for tensor product or Cartesian product
• RingElement ** ZZ, see ** -- a binary operator, usually used for tensor product or Cartesian product
• 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
• 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
• RingElement == RingElement, see == -- equality
• annihilator(RingElement), see annihilator -- the annihilator ideal
• baseName(RingElement), see baseName -- the base name of a generator
• coefficients(RingElement), see coefficients -- monomials and their coefficients
• cokernel(RingElement), see cokernel -- cokernel of a map of modules, graded modules, or chaincomplexes
• 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) -- 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
• 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(Vector,RingElement), see homogenize -- homogenize with respect to a variable
• homogenize(Vector,RingElement,List), see homogenize -- homogenize with respect to a variable
• ideal(RingElement) -- make an ideal
• image(RingElement), see image -- image of a map
• index(RingElement), see index -- numeric index of a ring variable
• indices(RingElement) -- indices of variables occuring in a polynomial
• isConstant(RingElement), see isConstant -- whether a ring element is constant
• isHomogeneous(RingElement), see isHomogeneous -- whether something is homogeneous (graded)
• isUnit(RingElement), see isUnit -- whether a ring element is a unit
• kernel(RingElement), see kernel(Matrix) -- kernel of a matrix
• lift(Ideal,type of RingElement), see lift -- lift to another ring
• lift(Matrix,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, 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, 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, see methods for normal forms and remainder -- calculate the normal form of ring elements and matrices
• RingElement % Ideal, see methods for normal forms and remainder -- calculate the normal form of ring elements and matrices
• RingElement % Matrix, see methods for normal forms and remainder -- calculate the normal form of ring elements and matrices
• RingElement % MonomialIdeal, see methods for normal forms and remainder -- calculate the normal form of ring elements and matrices
• Module / RingElement, see Module / Module -- quotient module
• monomialIdeal(RingElement), see monomialIdeal(Matrix) -- monomial ideal of lead terms
• monomials(RingElement), see monomials -- matrix of monomials in a ring element or matrix
• plethysm(List,RingElement) -- missing documentation
• plethysm(RingElement,RingElement) -- missing documentation
• 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
• 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 Array -- substitution of variables
• 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
• 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 occuring in a polynomial
• sylvesterMatrix(RingElement,RingElement,RingElement)
• 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
• weightRange(List,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.