# OrdinaryDifferentialRing(Kernels, R, var)ΒΆ

lodo.spad line 284 [edit on github]

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

- ^: (%, Integer) -> % if R has Field
from DivisionRing

- ^: (%, NonNegativeInteger) -> %
from MagmaWithUnit

- ^: (%, PositiveInteger) -> %
from Magma

- annihilate?: (%, %) -> Boolean
from Rng

- antiCommutator: (%, %) -> %

- 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
- 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

- gcdPolynomial: (SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> SparseUnivariatePolynomial % if R has Field
from GcdDomain

- inv: % -> % if R has Field
from DivisionRing

- latex: % -> String
from SetCategory

- 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

- plenaryPower: (%, PositiveInteger) -> % if R has Field
from NonAssociativeAlgebra %

- 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

- squareFreePart: % -> % if R has Field

- subtractIfCan: (%, %) -> Union(%, failed)

- 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

NonAssociativeAlgebra % if R has Field

NonAssociativeAlgebra Fraction Integer if R has Field

noZeroDivisors if R has Field

PrincipalIdealDomain if R has Field

RightModule Fraction Integer if R has Field

TwoSidedRecip if R has Field

UniqueFactorizationDomain if R has Field