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# NCRingElement -- exported from the NCAlgebra package, used to encode tensors

## Functions and methods returning an object of class NCRingElement:

- `CAxisTensor(ZZ,NCPolynomialRing)` -- see CAxisTensor -- the signature tensor of the canonical axis path at a given level
- `CMonTensor(ZZ,NCPolynomialRing)` -- see CMonTensor -- the signature tensor of the canonical monomial path at a given level
- `matrixAction(Matrix,NCRing,NCRing)` -- see matrixAction -- Diagonal matrix action on a tensor.
- `matrixAction(Matrix,NCRingElement,NCRing)` -- see matrixAction -- Diagonal matrix action on a tensor.

## Methods that use an object of class NCRingElement:

- `adjointWord(NCRingElement,NCPolynomialRing,List)` -- see adjointWord -- image of a word through the shuffle algebra homomorphism induced by a polynomial map
- antipode(NCRingElement) -- The antipode of a tensor
- baseName(NCRingElement) -- Returns the base name of a generator of an NCRing
- `coordinates(NCRingElement)` -- see coordinates -- Computes coordinates relative to a given basis
- degree(NCRingElement) -- Returns the degree of an NCRingElement
- `halfshuffle(NCRingElement,NCRingElement)` -- see halfshuffle -- compute the half-shuffle of an ordered pair of words
- `NCRingElement >> NCRingElement` -- see halfshuffle -- compute the half-shuffle of an ordered pair of words
- `inner(NCRingElement,NCRingElement)` -- see inner -- compute the inner product of two tensors.
- `NCRingElement @ NCRingElement` -- see inner -- compute the inner product of two tensors.
- `isCentral(NCRingElement)` -- see isCentral -- Determines if an element is central
- `isCentral(NCRingElement,NCGroebnerBasis)` -- see isCentral -- Determines if an element is central
- isConstant(NCRingElement) -- Returns whether the NCRingElement is constant
- `isHomogeneous(NCRingElement)` -- see isHomogeneous(NCIdeal) -- Determines whether the input defines a homogeneous object
- `isLeftRegular(NCRingElement,ZZ)` -- see isLeftRegular -- Determines if a given (homogeneous) element is regular in a given degree
- `isRightRegular(NCRingElement,ZZ)` -- see isLeftRegular -- Determines if a given (homogeneous) element is regular in a given degree
- isNormal(NCRingElement) -- Determines if a given NCRingElement is normal
- leadCoefficient(NCRingElement) -- Returns the lead monomial of an NCRingElement
- leadMonomial(NCRingElement) -- Returns the lead monomial of an NCRingElement
- leadTerm(NCRingElement) -- Returns the lead term of an NCRingElement
- `leftMultiplicationMap(NCRingElement,List,List)` -- see leftMultiplicationMap -- Computes a matrix for left or right multiplication by a homogeneous element
- `leftMultiplicationMap(NCRingElement,ZZ)` -- see leftMultiplicationMap -- Computes a matrix for left or right multiplication by a homogeneous element
- `leftMultiplicationMap(NCRingElement,ZZ,ZZ)` -- see leftMultiplicationMap -- Computes a matrix for left or right multiplication by a homogeneous element
- `rightMultiplicationMap(NCRingElement,List,List)` -- see leftMultiplicationMap -- Computes a matrix for left or right multiplication by a homogeneous element
- `rightMultiplicationMap(NCRingElement,ZZ)` -- see leftMultiplicationMap -- Computes a matrix for left or right multiplication by a homogeneous element
- `rightMultiplicationMap(NCRingElement,ZZ,ZZ)` -- see leftMultiplicationMap -- Computes a matrix for left or right multiplication by a homogeneous element
- List * NCRingElement -- Scales a list by an NCRingElement on the right
- `lyndonShuffle(NCRingElement)` -- see lyndonShuffle -- compute the representation of a tensor as a shuffle polynomial in Lyndon words
- `Matrix * NCRingElement` -- see matrixAction -- Diagonal matrix action on a tensor.
- monomials(NCRingElement) -- Returns the monomials appearing in the NCRingElement
- `ncIdeal(NCRingElement)` -- see ncIdeal -- Define a two-sided ideal in a noncommutative ring
- `ncLeftIdeal(NCRingElement)` -- see ncLeftIdeal -- Define a left ideal in a noncommutative ring
- NCMatrix * NCRingElement -- Product of NCMatrices
- `ncRightIdeal(NCRingElement)` -- see ncRightIdeal -- Define a right ideal in a noncommutative ring
- NCRingElement % NCGroebnerBasis -- Reduces a NCRingElement by a NCGroebnerBasis
- NCRingElement * List -- Scales a list by an NCRingElement on the left
- NCRingElement * NCMatrix -- Product of NCMatrices
- NCRingMap NCRingElement -- Apply an NCRingMap to an element or matrix
- `normalAutomorphism(NCRingElement)` -- see normalAutomorphism -- Computes the automorphism determined by a normal homogeneous element
- `normalFormBergman(NCRingElement,NCGroebnerBasis)` -- see normalFormBergman -- Calls Bergman for a normal form calculation
- `oreExtension(NCRing,NCRingMap,NCRingElement)` -- see oreExtension -- Creates an Ore extension of a noncommutative ring
- `oreExtension(NCRing,NCRingMap,NCRingMap,NCRingElement)` -- see oreExtension -- Creates an Ore extension of a noncommutative ring
- `oreIdeal(NCRing,NCRingMap,NCRingElement)` -- see oreIdeal -- Creates the defining ideal of an Ore extension of a noncommutative ring
- `oreIdeal(NCRing,NCRingMap,NCRingMap,NCRingElement)` -- see oreIdeal -- Creates the defining ideal of an Ore extension of a noncommutative ring
- ring(NCRingElement) -- Returns the NCRing of an NCRingElement
- `NCRingElement ** NCRingElement` -- see shuffle -- shuffle product of two words
- `NCRingElement ⧢ NCRingElement` -- see shuffle -- shuffle product of two words
- `shuffle(NCRingElement,NCRingElement)` -- see shuffle -- shuffle product of two words
- `sig(Path,NCRingElement)` -- see sig -- compute the signature of a piecewise polynomial path.
- size(NCRingElement) -- Returns the number of terms of an NCRingElement
- `sparseCoeffs(NCRingElement)` -- see sparseCoeffs -- Converts ring elements into vectors over the coefficient ring
- support(NCRingElement) -- Returns the variables appearing in the NCRingElement
- `NCRingElement @ ZZ` -- see tensorArray -- k-th level component of a tensor.
- `tensorArray(NCRingElement)` -- see tensorArray -- k-th level component of a tensor.
- `tensorArray(NCRingElement,ZZ)` -- see tensorArray -- k-th level component of a tensor.
- `tensorExp(NCRingElement,ZZ)` -- see tensorExp -- Compute a component of the exponential of a tensor.
- `tensorLog(NCRingElement,ZZ)` -- see tensorLog -- Compute a component of the logarithm of a tensor.
- `tensorParametrization(NCRingElement)` -- see tensorParametrization -- Constructs the morphism that maps a word to its coefficient in the given tensor.
- terms(NCRingElement) -- Returns the terms of an NCRingElement
- toString(NCRingElement) -- Converts an NCRingElement to a string
- `wordFormat(NCRingElement)` -- see wordFormat -- display a tensor in word notation
- `wordString(NCRingElement)` -- see wordString -- a string representing a word in wordFormat

## For the programmer

The object NCRingElement is a type, with ancestor classes HashTable < Thing.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/PathSignatures/documentation.m2:159:0.