# OrdinaryDifferentialRing(Kernels, R, var)ΒΆ

- Kernels: SetCategory
- R: PartialDifferentialRing Kernels
- var: Kernels

This constructor produces an ordinary differential ring from a partial differential ring by specifying a variable.

- 0: %
- from AbelianMonoid
- 1: %
- from MagmaWithUnit
- *: (%, %) -> %
- from LeftModule %
- *: (%, Fraction Integer) -> % if R has Field
- from RightModule Fraction Integer
- *: (Fraction Integer, %) -> % if R has Field
- from LeftModule Fraction Integer
- *: (Integer, %) -> %
- from AbelianGroup
- *: (NonNegativeInteger, %) -> %
- from AbelianMonoid
- *: (PositiveInteger, %) -> %
- from AbelianSemiGroup
- +: (%, %) -> %
- from AbelianSemiGroup
- -: % -> %
- from AbelianGroup
- -: (%, %) -> %
- from AbelianGroup
- /: (%, %) -> % if R has Field
- from Field
- =: (%, %) -> Boolean
- from BasicType
- ^: (%, Integer) -> % if R has Field
- from DivisionRing
- ^: (%, NonNegativeInteger) -> %
- from MagmaWithUnit
- ^: (%, PositiveInteger) -> %
- from Magma
- ~=: (%, %) -> Boolean
- from BasicType
- annihilate?: (%, %) -> Boolean
- from Rng
- antiCommutator: (%, %) -> %
- from NonAssociativeSemiRng
- associates?: (%, %) -> Boolean if R has Field
- from EntireRing
- associator: (%, %, %) -> %
- from NonAssociativeRng
- characteristic: () -> NonNegativeInteger
- from NonAssociativeRing
- coerce: % -> % if R has Field
- from Algebra %
- coerce: % -> OutputForm
- from CoercibleTo OutputForm

- coerce: % -> R
`coerce(p)`

views`p`

as a valie in the partial differential ring.- coerce: Fraction Integer -> % if R has Field
- from Algebra Fraction Integer
- coerce: Integer -> %
- from NonAssociativeRing

- coerce: R -> %
`coerce(r)`

views`r`

as a value in the ordinary differential ring.- commutator: (%, %) -> %
- from NonAssociativeRng
- D: % -> %
- from DifferentialRing
- D: (%, NonNegativeInteger) -> %
- from DifferentialRing
- differentiate: % -> %
- from DifferentialRing
- differentiate: (%, NonNegativeInteger) -> %
- from DifferentialRing
- divide: (%, %) -> Record(quotient: %, remainder: %) if R has Field
- from EuclideanDomain
- euclideanSize: % -> NonNegativeInteger if R has Field
- from EuclideanDomain
- expressIdealMember: (List %, %) -> Union(List %, failed) if R has Field
- from PrincipalIdealDomain
- exquo: (%, %) -> Union(%, failed) if R has Field
- from EntireRing
- extendedEuclidean: (%, %) -> Record(coef1: %, coef2: %, generator: %) if R has Field
- from EuclideanDomain
- extendedEuclidean: (%, %, %) -> Union(Record(coef1: %, coef2: %), failed) if R has Field
- from EuclideanDomain
- factor: % -> Factored % if R has Field
- from UniqueFactorizationDomain
- gcd: (%, %) -> % if R has Field
- from GcdDomain
- gcd: List % -> % if R has Field
- from GcdDomain
- gcdPolynomial: (SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> SparseUnivariatePolynomial % if R has Field
- from GcdDomain
- hash: % -> SingleInteger
- from SetCategory
- hashUpdate!: (HashState, %) -> HashState
- from SetCategory
- inv: % -> % if R has Field
- from DivisionRing
- latex: % -> String
- from SetCategory
- lcm: (%, %) -> % if R has Field
- from GcdDomain
- lcm: List % -> % if R has Field
- from GcdDomain
- lcmCoef: (%, %) -> Record(llcm_res: %, coeff1: %, coeff2: %) if R has Field
- from LeftOreRing
- leftPower: (%, NonNegativeInteger) -> %
- from MagmaWithUnit
- leftPower: (%, PositiveInteger) -> %
- from Magma
- leftRecip: % -> Union(%, failed)
- from MagmaWithUnit
- multiEuclidean: (List %, %) -> Union(List %, failed) if R has Field
- from EuclideanDomain
- one?: % -> Boolean
- from MagmaWithUnit
- opposite?: (%, %) -> Boolean
- from AbelianMonoid
- prime?: % -> Boolean if R has Field
- from UniqueFactorizationDomain
- principalIdeal: List % -> Record(coef: List %, generator: %) if R has Field
- from PrincipalIdealDomain
- quo: (%, %) -> % if R has Field
- from EuclideanDomain
- recip: % -> Union(%, failed)
- from MagmaWithUnit
- rem: (%, %) -> % if R has Field
- from EuclideanDomain
- rightPower: (%, NonNegativeInteger) -> %
- from MagmaWithUnit
- rightPower: (%, PositiveInteger) -> %
- from Magma
- rightRecip: % -> Union(%, failed)
- from MagmaWithUnit
- sample: %
- from AbelianMonoid
- sizeLess?: (%, %) -> Boolean if R has Field
- from EuclideanDomain
- squareFree: % -> Factored % if R has Field
- from UniqueFactorizationDomain
- squareFreePart: % -> % if R has Field
- from UniqueFactorizationDomain
- subtractIfCan: (%, %) -> Union(%, failed)
- from CancellationAbelianMonoid
- unit?: % -> Boolean if R has Field
- from EntireRing
- unitCanonical: % -> % if R has Field
- from EntireRing
- unitNormal: % -> Record(unit: %, canonical: %, associate: %) if R has Field
- from EntireRing
- zero?: % -> Boolean
- from AbelianMonoid

Algebra Fraction Integer if R has Field

BiModule(%, %)

BiModule(Fraction Integer, Fraction Integer) if R has Field

canonicalsClosed if R has Field

canonicalUnitNormal if R has Field

CommutativeRing if R has Field

CommutativeStar if R has Field

DivisionRing if R has Field

EntireRing if R has Field

EuclideanDomain if R has Field

IntegralDomain if R has Field

LeftModule Fraction Integer if R has Field

LeftOreRing if R has Field

Module Fraction Integer if R has Field

noZeroDivisors if R has Field

PrincipalIdealDomain if R has Field

RightModule Fraction Integer if R has Field

UniqueFactorizationDomain if R has Field