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

## See also

- EngineRing -- the class of rings handled by the engine

### Menu

- isUnit -- whether a ring element is a unit
- leadTerm(RingElement) -- get the greatest term
- leadTerm(ZZ,RingElement) -- get the lead polynomials using part of the monomial order
- quotientRemainder(RingElement,RingElement) -- quotient and remainder
- degree(RingElement)
- degree(RingElement,RingElement) -- degree with respect to a variable
- factor(RingElement) -- factor a ring element
- indices(RingElement) -- indices of variables occurring in a polynomial
- RingElement ^ ZZ -- power
- RingElement / RingElement -- fraction
- RingElement .. RingElement -- a sequence of consecutive generators of a polynomial ring
- RingElement ..< RingElement -- a sequence of consecutive generators of a polynomial ring

## 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)
- `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 /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Macaulay2Doc/doc_rings.m2:177:0.