Complex R

gaussian.spad line 542 [edit on github]

• R: CommutativeRing

spadtype {Complex(R)} creates the domain of elements of the form a + b * i where a and b come from the ring R, and i is a new element such that i^2 = -1.

0: %

from AbelianMonoid

1: %

from MagmaWithUnit

*: (%, %) -> %

from LeftModule %

*: (%, Fraction Integer) -> % if R has Field

from RightModule Fraction Integer

*: (%, Integer) -> % if R has LinearlyExplicitOver Integer

from RightModule Integer

*: (%, R) -> %

from RightModule R

*: (Fraction Integer, %) -> % if R has Field

from LeftModule Fraction Integer

*: (Integer, %) -> %

from AbelianGroup

*: (NonNegativeInteger, %) -> %

from AbelianMonoid

*: (PositiveInteger, %) -> %

from AbelianSemiGroup

*: (R, %) -> %

from LeftModule R

+: (%, %) -> %

from AbelianSemiGroup

-: % -> %

from AbelianGroup

-: (%, %) -> %

from AbelianGroup

/: (%, %) -> % if R has Field

from Field

=: (%, %) -> Boolean

from BasicType

^: (%, %) -> % if R has TranscendentalFunctionCategory

from ElementaryFunctionCategory

^: (%, Fraction Integer) -> % if R has RadicalCategory and R has TranscendentalFunctionCategory

from RadicalCategory

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

from DivisionRing

^: (%, NonNegativeInteger) -> %

from MagmaWithUnit

^: (%, PositiveInteger) -> %

from Magma

~=: (%, %) -> Boolean

from BasicType

abs: % -> % if R has RealNumberSystem

from ComplexCategory R

acos: % -> % if R has TranscendentalFunctionCategory

from ArcTrigonometricFunctionCategory

acosh: % -> % if R has TranscendentalFunctionCategory

from ArcHyperbolicFunctionCategory

acot: % -> % if R has TranscendentalFunctionCategory

from ArcTrigonometricFunctionCategory

acoth: % -> % if R has TranscendentalFunctionCategory

from ArcHyperbolicFunctionCategory

acsc: % -> % if R has TranscendentalFunctionCategory

from ArcTrigonometricFunctionCategory

acsch: % -> % if R has TranscendentalFunctionCategory

from ArcHyperbolicFunctionCategory

annihilate?: (%, %) -> Boolean

from Rng

antiCommutator: (%, %) -> %

from NonAssociativeSemiRng

argument: % -> R if R has TranscendentalFunctionCategory

from ComplexCategory R

asec: % -> % if R has TranscendentalFunctionCategory

from ArcTrigonometricFunctionCategory

asech: % -> % if R has TranscendentalFunctionCategory

from ArcHyperbolicFunctionCategory

asin: % -> % if R has TranscendentalFunctionCategory

from ArcTrigonometricFunctionCategory

asinh: % -> % if R has TranscendentalFunctionCategory

from ArcHyperbolicFunctionCategory

associates?: (%, %) -> Boolean if R has IntegralDomain

from EntireRing

associator: (%, %, %) -> %

from NonAssociativeRng

atan: % -> % if R has TranscendentalFunctionCategory

from ArcTrigonometricFunctionCategory

atanh: % -> % if R has TranscendentalFunctionCategory

from ArcHyperbolicFunctionCategory

basis: () -> Vector %

from FramedModule R

characteristic: () -> NonNegativeInteger

from NonAssociativeRing

characteristicPolynomial: % -> SparseUnivariatePolynomial R

from FiniteRankAlgebra(R, SparseUnivariatePolynomial R)

charthRoot: % -> % if R has FiniteFieldCategory

from FiniteFieldCategory

charthRoot: % -> Union(%, failed) if R has EuclideanDomain and % has CharacteristicNonZero and R has PolynomialFactorizationExplicit or R has CharacteristicNonZero

from CharacteristicNonZero

coerce: % -> %

from Algebra %

coerce: % -> OutputForm

from CoercibleTo OutputForm

coerce: Fraction Integer -> % if R has Field or R has RetractableTo Fraction Integer

from Algebra Fraction Integer

coerce: Integer -> %

from NonAssociativeRing

coerce: R -> %

from Algebra R

commutator: (%, %) -> %

from NonAssociativeRng

complex: (R, R) -> %

from ComplexCategory R

conditionP: Matrix % -> Union(Vector %, failed) if R has EuclideanDomain and % has CharacteristicNonZero and R has PolynomialFactorizationExplicit or R has FiniteFieldCategory

from PolynomialFactorizationExplicit

conjugate: % -> %

from ComplexCategory R

convert: % -> Complex DoubleFloat if R has RealConstant

from ConvertibleTo Complex DoubleFloat

convert: % -> Complex Float if R has RealConstant

from ConvertibleTo Complex Float

convert: % -> InputForm if R has ConvertibleTo InputForm

from ConvertibleTo InputForm

convert: % -> Pattern Float if R has ConvertibleTo Pattern Float

from ConvertibleTo Pattern Float

convert: % -> Pattern Integer if R has ConvertibleTo Pattern Integer

from ConvertibleTo Pattern Integer

convert: % -> SparseUnivariatePolynomial R

from ConvertibleTo SparseUnivariatePolynomial R

convert: % -> Vector R

from FramedModule R

convert: SparseUnivariatePolynomial R -> %

from MonogenicAlgebra(R, SparseUnivariatePolynomial R)

convert: Vector R -> %

from FramedModule R

coordinates: % -> Vector R

from FramedModule R

coordinates: (%, Vector %) -> Vector R

from FiniteRankAlgebra(R, SparseUnivariatePolynomial R)

coordinates: (Vector %, Vector %) -> Matrix R

from FiniteRankAlgebra(R, SparseUnivariatePolynomial R)

coordinates: Vector % -> Matrix R

from FramedModule R

cos: % -> % if R has TranscendentalFunctionCategory

from TrigonometricFunctionCategory

cosh: % -> % if R has TranscendentalFunctionCategory

from HyperbolicFunctionCategory

cot: % -> % if R has TranscendentalFunctionCategory

from TrigonometricFunctionCategory

coth: % -> % if R has TranscendentalFunctionCategory

from HyperbolicFunctionCategory

createPrimitiveElement: () -> % if R has FiniteFieldCategory

from FiniteFieldCategory

csc: % -> % if R has TranscendentalFunctionCategory

from TrigonometricFunctionCategory

csch: % -> % if R has TranscendentalFunctionCategory

from HyperbolicFunctionCategory

D: % -> % if R has DifferentialRing

from DifferentialRing

D: (%, List Symbol) -> % if R has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

D: (%, List Symbol, List NonNegativeInteger) -> % if R has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

D: (%, NonNegativeInteger) -> % if R has DifferentialRing

from DifferentialRing

D: (%, R -> R) -> %

from DifferentialExtension R

D: (%, R -> R, NonNegativeInteger) -> %

from DifferentialExtension R

D: (%, Symbol) -> % if R has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

D: (%, Symbol, NonNegativeInteger) -> % if R has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

definingPolynomial: () -> SparseUnivariatePolynomial R

from MonogenicAlgebra(R, SparseUnivariatePolynomial R)

derivationCoordinates: (Vector %, R -> R) -> Matrix R if R has Field

from MonogenicAlgebra(R, SparseUnivariatePolynomial R)

differentiate: % -> % if R has DifferentialRing

from DifferentialRing

differentiate: (%, List Symbol) -> % if R has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

differentiate: (%, List Symbol, List NonNegativeInteger) -> % if R has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

differentiate: (%, NonNegativeInteger) -> % if R has DifferentialRing

from DifferentialRing

differentiate: (%, R -> R) -> %

from DifferentialExtension R

differentiate: (%, R -> R, NonNegativeInteger) -> %

from DifferentialExtension R

differentiate: (%, Symbol) -> % if R has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

differentiate: (%, Symbol, NonNegativeInteger) -> % if R has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

discreteLog: % -> NonNegativeInteger if R has FiniteFieldCategory

from FiniteFieldCategory

discreteLog: (%, %) -> Union(NonNegativeInteger, failed) if R has FiniteFieldCategory

from FieldOfPrimeCharacteristic

discriminant: () -> R

from FramedAlgebra(R, SparseUnivariatePolynomial R)

discriminant: Vector % -> R

from FiniteRankAlgebra(R, SparseUnivariatePolynomial R)

divide: (%, %) -> Record(quotient: %, remainder: %) if R has Field or R has IntegerNumberSystem

from EuclideanDomain

elt: (%, R) -> % if R has Eltable(R, R)

from Eltable(R, %)

enumerate: () -> List % if R has Finite

from Finite

euclideanSize: % -> NonNegativeInteger if R has Field or R has IntegerNumberSystem

from EuclideanDomain

eval: (%, Equation R) -> % if R has Evalable R

from Evalable R

eval: (%, List Equation R) -> % if R has Evalable R

from Evalable R

eval: (%, List R, List R) -> % if R has Evalable R

from InnerEvalable(R, R)

eval: (%, List Symbol, List R) -> % if R has InnerEvalable(Symbol, R)

from InnerEvalable(Symbol, R)

eval: (%, R, R) -> % if R has Evalable R

from InnerEvalable(R, R)

eval: (%, Symbol, R) -> % if R has InnerEvalable(Symbol, R)

from InnerEvalable(Symbol, R)

exp: % -> % if R has TranscendentalFunctionCategory

from ElementaryFunctionCategory

expressIdealMember: (List %, %) -> Union(List %, failed) if R has Field or R has IntegerNumberSystem

from PrincipalIdealDomain

exquo: (%, %) -> Union(%, failed) if R has IntegralDomain

from EntireRing

exquo: (%, R) -> Union(%, failed) if R has IntegralDomain

from ComplexCategory R

extendedEuclidean: (%, %) -> Record(coef1: %, coef2: %, generator: %) if R has Field or R has IntegerNumberSystem

from EuclideanDomain

extendedEuclidean: (%, %, %) -> Union(Record(coef1: %, coef2: %), failed) if R has Field or R has IntegerNumberSystem

from EuclideanDomain

factor: % -> Factored % if R has IntegerNumberSystem or R has EuclideanDomain and R has PolynomialFactorizationExplicit or R has Field

from UniqueFactorizationDomain

factorPolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if R has EuclideanDomain and R has PolynomialFactorizationExplicit or R has FiniteFieldCategory

from PolynomialFactorizationExplicit

factorsOfCyclicGroupSize: () -> List Record(factor: Integer, exponent: NonNegativeInteger) if R has FiniteFieldCategory

from FiniteFieldCategory

factorSquareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if R has EuclideanDomain and R has PolynomialFactorizationExplicit or R has FiniteFieldCategory

from PolynomialFactorizationExplicit

gcd: (%, %) -> % if R has IntegerNumberSystem or R has EuclideanDomain and R has PolynomialFactorizationExplicit or R has Field

from GcdDomain

gcd: List % -> % if R has IntegerNumberSystem or R has EuclideanDomain and R has PolynomialFactorizationExplicit or R has Field

from GcdDomain

gcdPolynomial: (SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> SparseUnivariatePolynomial % if R has IntegerNumberSystem or R has EuclideanDomain and R has PolynomialFactorizationExplicit or R has Field

from GcdDomain

generator: () -> %

from MonogenicAlgebra(R, SparseUnivariatePolynomial R)

hash: % -> SingleInteger if R has Hashable

from Hashable

hashUpdate!: (HashState, %) -> HashState if R has Hashable

from Hashable

imag: % -> R

from ComplexCategory R

imaginary: () -> %

from ComplexCategory R

index: PositiveInteger -> % if R has Finite

from Finite

init: % if R has FiniteFieldCategory

from StepThrough

inv: % -> % if R has Field

from DivisionRing

latex: % -> String

from SetCategory

lcm: (%, %) -> % if R has IntegerNumberSystem or R has EuclideanDomain and R has PolynomialFactorizationExplicit or R has Field

from GcdDomain

lcm: List % -> % if R has IntegerNumberSystem or R has EuclideanDomain and R has PolynomialFactorizationExplicit or R has Field

from GcdDomain

lcmCoef: (%, %) -> Record(llcm_res: %, coeff1: %, coeff2: %) if R has IntegerNumberSystem or R has EuclideanDomain and R has PolynomialFactorizationExplicit or R has Field

from LeftOreRing

leftPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

leftPower: (%, PositiveInteger) -> %

from Magma

leftRecip: % -> Union(%, failed)

from MagmaWithUnit

lift: % -> SparseUnivariatePolynomial R

from MonogenicAlgebra(R, SparseUnivariatePolynomial R)

log: % -> % if R has TranscendentalFunctionCategory

from ElementaryFunctionCategory

lookup: % -> PositiveInteger if R has Finite

from Finite

map: (R -> R, %) -> %

from FullyEvalableOver R

minimalPolynomial: % -> SparseUnivariatePolynomial R if R has Field

from FiniteRankAlgebra(R, SparseUnivariatePolynomial R)

multiEuclidean: (List %, %) -> Union(List %, failed) if R has Field or R has IntegerNumberSystem

from EuclideanDomain

nextItem: % -> Union(%, failed) if R has FiniteFieldCategory

from StepThrough

norm: % -> R

from ComplexCategory R

nthRoot: (%, Integer) -> % if R has RadicalCategory and R has TranscendentalFunctionCategory

from RadicalCategory

OMwrite: % -> String if R has OpenMath

from OpenMath

OMwrite: (%, Boolean) -> String if R has OpenMath

from OpenMath

OMwrite: (OpenMathDevice, %) -> Void if R has OpenMath

from OpenMath

OMwrite: (OpenMathDevice, %, Boolean) -> Void if R has OpenMath

from OpenMath

one?: % -> Boolean

from MagmaWithUnit

opposite?: (%, %) -> Boolean

from AbelianMonoid

order: % -> OnePointCompletion PositiveInteger if R has FiniteFieldCategory

from FieldOfPrimeCharacteristic

order: % -> PositiveInteger if R has FiniteFieldCategory

from FiniteFieldCategory

patternMatch: (%, Pattern Float, PatternMatchResult(Float, %)) -> PatternMatchResult(Float, %) if R has PatternMatchable Float

from PatternMatchable Float

patternMatch: (%, Pattern Integer, PatternMatchResult(Integer, %)) -> PatternMatchResult(Integer, %) if R has PatternMatchable Integer

from PatternMatchable Integer

pi: () -> % if R has TranscendentalFunctionCategory

from TranscendentalFunctionCategory

plenaryPower: (%, PositiveInteger) -> %

from NonAssociativeAlgebra %

polarCoordinates: % -> Record(r: R, phi: R) if R has RealNumberSystem and R has TranscendentalFunctionCategory

from ComplexCategory R

prime?: % -> Boolean if R has IntegerNumberSystem or R has EuclideanDomain and R has PolynomialFactorizationExplicit or R has Field

from UniqueFactorizationDomain

primeFrobenius: % -> % if R has FiniteFieldCategory

from FieldOfPrimeCharacteristic

primeFrobenius: (%, NonNegativeInteger) -> % if R has FiniteFieldCategory

from FieldOfPrimeCharacteristic

primitive?: % -> Boolean if R has FiniteFieldCategory

from FiniteFieldCategory

primitiveElement: () -> % if R has FiniteFieldCategory

from FiniteFieldCategory

principalIdeal: List % -> Record(coef: List %, generator: %) if R has Field or R has IntegerNumberSystem

from PrincipalIdealDomain

quo: (%, %) -> % if R has Field or R has IntegerNumberSystem

from EuclideanDomain

random: () -> % if R has Finite

from Finite

rank: () -> PositiveInteger

from FramedModule R

rational?: % -> Boolean if R has IntegerNumberSystem

from ComplexCategory R

rational: % -> Fraction Integer if R has IntegerNumberSystem

from ComplexCategory R

rationalIfCan: % -> Union(Fraction Integer, failed) if R has IntegerNumberSystem

from ComplexCategory R

real: % -> R

from ComplexCategory R

recip: % -> Union(%, failed)

from MagmaWithUnit

reduce: Fraction SparseUnivariatePolynomial R -> Union(%, failed) if R has Field

from MonogenicAlgebra(R, SparseUnivariatePolynomial R)

reduce: SparseUnivariatePolynomial R -> %

from MonogenicAlgebra(R, SparseUnivariatePolynomial R)

reducedSystem: (Matrix %, Vector %) -> Record(mat: Matrix Integer, vec: Vector Integer) if R has LinearlyExplicitOver Integer

from LinearlyExplicitOver Integer

reducedSystem: (Matrix %, Vector %) -> Record(mat: Matrix R, vec: Vector R)

from LinearlyExplicitOver R

reducedSystem: Matrix % -> Matrix Integer if R has LinearlyExplicitOver Integer

from LinearlyExplicitOver Integer

reducedSystem: Matrix % -> Matrix R

from LinearlyExplicitOver R

regularRepresentation: % -> Matrix R

from FramedAlgebra(R, SparseUnivariatePolynomial R)

regularRepresentation: (%, Vector %) -> Matrix R

from FiniteRankAlgebra(R, SparseUnivariatePolynomial R)

rem: (%, %) -> % if R has Field or R has IntegerNumberSystem

from EuclideanDomain

representationType: () -> Union(prime, polynomial, normal, cyclic) if R has FiniteFieldCategory

from FiniteFieldCategory

represents: (Vector R, Vector %) -> %

from FiniteRankAlgebra(R, SparseUnivariatePolynomial R)

represents: Vector R -> %

from FramedModule R

retract: % -> Fraction Integer if R has RetractableTo Fraction Integer

from RetractableTo Fraction Integer

retract: % -> Integer if R has RetractableTo Integer

from RetractableTo Integer

retract: % -> R

from RetractableTo R

retractIfCan: % -> Union(Fraction Integer, failed) if R has RetractableTo Fraction Integer

from RetractableTo Fraction Integer

retractIfCan: % -> Union(Integer, failed) if R has RetractableTo Integer

from RetractableTo Integer

retractIfCan: % -> Union(R, failed)

from RetractableTo R

rightPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower: (%, PositiveInteger) -> %

from Magma

rightRecip: % -> Union(%, failed)

from MagmaWithUnit

sample: %

from AbelianMonoid

sec: % -> % if R has TranscendentalFunctionCategory

from TrigonometricFunctionCategory

sech: % -> % if R has TranscendentalFunctionCategory

from HyperbolicFunctionCategory

sin: % -> % if R has TranscendentalFunctionCategory

from TrigonometricFunctionCategory

sinh: % -> % if R has TranscendentalFunctionCategory

from HyperbolicFunctionCategory

size: () -> NonNegativeInteger if R has Finite

from Finite

sizeLess?: (%, %) -> Boolean if R has Field or R has IntegerNumberSystem

from EuclideanDomain

smaller?: (%, %) -> Boolean if R has Comparable

from Comparable

solveLinearPolynomialEquation: (List SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> Union(List SparseUnivariatePolynomial %, failed) if R has EuclideanDomain and R has PolynomialFactorizationExplicit or R has FiniteFieldCategory

from PolynomialFactorizationExplicit

sqrt: % -> % if R has RadicalCategory and R has TranscendentalFunctionCategory

from RadicalCategory

squareFree: % -> Factored % if R has IntegerNumberSystem or R has EuclideanDomain and R has PolynomialFactorizationExplicit or R has Field

from UniqueFactorizationDomain

squareFreePart: % -> % if R has IntegerNumberSystem or R has EuclideanDomain and R has PolynomialFactorizationExplicit or R has Field

from UniqueFactorizationDomain

squareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if R has EuclideanDomain and R has PolynomialFactorizationExplicit or R has FiniteFieldCategory

from PolynomialFactorizationExplicit

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

from CancellationAbelianMonoid

tableForDiscreteLogarithm: Integer -> Table(PositiveInteger, NonNegativeInteger) if R has FiniteFieldCategory

from FiniteFieldCategory

tan: % -> % if R has TranscendentalFunctionCategory

from TrigonometricFunctionCategory

tanh: % -> % if R has TranscendentalFunctionCategory

from HyperbolicFunctionCategory

trace: % -> R

from FiniteRankAlgebra(R, SparseUnivariatePolynomial R)

traceMatrix: () -> Matrix R

from FramedAlgebra(R, SparseUnivariatePolynomial R)

traceMatrix: Vector % -> Matrix R

from FiniteRankAlgebra(R, SparseUnivariatePolynomial R)

unit?: % -> Boolean if R has IntegralDomain

from EntireRing

unitCanonical: % -> % if R has IntegralDomain

from EntireRing

unitNormal: % -> Record(unit: %, canonical: %, associate: %) if R has IntegralDomain

from EntireRing

zero?: % -> Boolean

from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianSemiGroup

Algebra %

Algebra Fraction Integer if R has Field

Algebra R

arbitraryPrecision if R has arbitraryPrecision

ArcHyperbolicFunctionCategory if R has TranscendentalFunctionCategory

ArcTrigonometricFunctionCategory if R has TranscendentalFunctionCategory

BasicType

BiModule(%, %)

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

BiModule(R, R)

CancellationAbelianMonoid

canonicalsClosed if R has Field

canonicalUnitNormal if R has Field

CharacteristicNonZero if R has CharacteristicNonZero

CharacteristicZero if R has CharacteristicZero

CoercibleFrom Fraction Integer if R has RetractableTo Fraction Integer

CoercibleFrom Integer if R has RetractableTo Integer

CoercibleFrom R

CoercibleTo OutputForm

CommutativeRing

CommutativeStar

Comparable if R has Comparable

ComplexCategory R

ConvertibleTo Complex DoubleFloat if R has RealConstant

ConvertibleTo Complex Float if R has RealConstant

ConvertibleTo InputForm if R has ConvertibleTo InputForm

ConvertibleTo Pattern Float if R has ConvertibleTo Pattern Float

ConvertibleTo Pattern Integer if R has ConvertibleTo Pattern Integer

ConvertibleTo SparseUnivariatePolynomial R

DifferentialExtension R

DifferentialRing if R has DifferentialRing

DivisionRing if R has Field

ElementaryFunctionCategory if R has TranscendentalFunctionCategory

Eltable(R, %) if R has Eltable(R, R)

EntireRing if R has IntegralDomain

EuclideanDomain if R has Field or R has IntegerNumberSystem

Evalable R if R has Evalable R

Field if R has Field

FieldOfPrimeCharacteristic if R has FiniteFieldCategory

Finite if R has Finite

FiniteFieldCategory if R has FiniteFieldCategory

FiniteRankAlgebra(R, SparseUnivariatePolynomial R)

FramedAlgebra(R, SparseUnivariatePolynomial R)

FramedModule R

FullyEvalableOver R

FullyLinearlyExplicitOver R

FullyPatternMatchable R

FullyRetractableTo R

GcdDomain if R has IntegerNumberSystem or R has EuclideanDomain and R has PolynomialFactorizationExplicit or R has Field

Hashable if R has Hashable

HyperbolicFunctionCategory if R has TranscendentalFunctionCategory

InnerEvalable(R, R) if R has Evalable R

InnerEvalable(Symbol, R) if R has InnerEvalable(Symbol, R)

IntegralDomain if R has IntegralDomain

LeftModule %

LeftModule Fraction Integer if R has Field

LeftModule R

LeftOreRing if R has IntegerNumberSystem or R has EuclideanDomain and R has PolynomialFactorizationExplicit or R has Field

LinearlyExplicitOver Integer if R has LinearlyExplicitOver Integer

LinearlyExplicitOver R

Magma

MagmaWithUnit

Module %

Module Fraction Integer if R has Field

Module R

MonogenicAlgebra(R, SparseUnivariatePolynomial R)

Monoid

multiplicativeValuation if R has IntegerNumberSystem

NonAssociativeAlgebra %

NonAssociativeAlgebra Fraction Integer if R has Field

NonAssociativeAlgebra R

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors if R has IntegralDomain

OpenMath if R has OpenMath

PartialDifferentialRing Symbol if R has PartialDifferentialRing Symbol

Patternable R

PatternMatchable Float if R has PatternMatchable Float

PatternMatchable Integer if R has PatternMatchable Integer

PolynomialFactorizationExplicit if R has EuclideanDomain and R has PolynomialFactorizationExplicit or R has FiniteFieldCategory

PrincipalIdealDomain if R has Field or R has IntegerNumberSystem

RadicalCategory if R has RadicalCategory and R has TranscendentalFunctionCategory

RetractableTo Fraction Integer if R has RetractableTo Fraction Integer

RetractableTo Integer if R has RetractableTo Integer

RetractableTo R

RightModule %

RightModule Fraction Integer if R has Field

RightModule Integer if R has LinearlyExplicitOver Integer

RightModule R

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

StepThrough if R has FiniteFieldCategory

TranscendentalFunctionCategory if R has TranscendentalFunctionCategory

TrigonometricFunctionCategory if R has TranscendentalFunctionCategory

TwoSidedRecip

UniqueFactorizationDomain if R has IntegerNumberSystem or R has EuclideanDomain and R has PolynomialFactorizationExplicit or R has Field

unitsKnown