PrincipalIdealDomain

catdef.spad line 1301 [edit on github]

The category of constructive principal ideal domains, i.e. where a single generator can be constructively found for any ideal given by a finite set of generators. Note that this constructive definition only implies that finitely generated ideals are principal. It is not clear what we would mean by an infinitely generated ideal.

* : (%, %) -> %

from Magma

* : (Integer, %) -> %

from AbelianGroup

* : (NonNegativeInteger, %) -> %

from AbelianMonoid

* : (PositiveInteger, %) -> %

from AbelianSemiGroup

+ : (%, %) -> %

from AbelianSemiGroup

- : % -> %

from AbelianGroup

- : (%, %) -> %

from AbelianGroup

0 : () -> %

from AbelianMonoid

1 : () -> %

from MagmaWithUnit

= : (%, %) -> Boolean

from BasicType

^ : (%, NonNegativeInteger) -> %

from MagmaWithUnit

^ : (%, PositiveInteger) -> %

from Magma

annihilate? : (%, %) -> Boolean

from Rng

antiCommutator : (%, %) -> %

from NonAssociativeSemiRng

associates? : (%, %) -> Boolean

from EntireRing

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

from NonAssociativeRng

characteristic : () -> NonNegativeInteger

from NonAssociativeRing

coerce : % -> %

from Algebra(%)

coerce : Integer -> %

from NonAssociativeRing

coerce : % -> OutputForm

from CoercibleTo(OutputForm)

commutator : (%, %) -> %

from NonAssociativeRng

expressIdealMember : (List(%), %) -> Union(List(%), "failed")

expressIdealMember([f1, ..., fn], h) returns a representation of h as a linear combination of the fi or "failed" if h is not in the ideal generated by the fi.

exquo : (%, %) -> Union(%, "failed")

from EntireRing

gcd : (%, %) -> %

from GcdDomain

gcd : List(%) -> %

from GcdDomain

gcdPolynomial : (SparseUnivariatePolynomial(%), SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%)

from GcdDomain

latex : % -> String

from SetCategory

lcm : (%, %) -> %

from GcdDomain

lcm : List(%) -> %

from GcdDomain

lcmCoef : (%, %) -> Record(llcm_res : %, coeff1 : %, coeff2 : %)

from LeftOreRing

leftPower : (%, NonNegativeInteger) -> %

from MagmaWithUnit

leftPower : (%, PositiveInteger) -> %

from Magma

leftRecip : % -> Union(%, "failed")

from MagmaWithUnit

one? : % -> Boolean

from MagmaWithUnit

opposite? : (%, %) -> Boolean

from AbelianMonoid

plenaryPower : (%, PositiveInteger) -> %

from NonAssociativeAlgebra(%)

principalIdeal : List(%) -> Record(coef : List(%), generator : %)

principalIdeal([f1, ..., fn]) returns a record whose generator component is a generator of the ideal generated by [f1, ..., fn] whose coef component satisfies generator = sum (input.i * coef.i)

recip : % -> Union(%, "failed")

from MagmaWithUnit

rightPower : (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower : (%, PositiveInteger) -> %

from Magma

rightRecip : % -> Union(%, "failed")

from MagmaWithUnit

sample : () -> %

from AbelianMonoid

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

from CancellationAbelianMonoid

unit? : % -> Boolean

from EntireRing

unitCanonical : % -> %

from EntireRing

unitNormal : % -> Record(unit : %, canonical : %, associate : %)

from EntireRing

zero? : % -> Boolean

from AbelianMonoid

~= : (%, %) -> Boolean

from BasicType

IntegralDomain

noZeroDivisors

NonAssociativeSemiRng

Algebra(%)

RightModule(%)

Monoid

GcdDomain

AbelianMonoid

CancellationAbelianMonoid

MagmaWithUnit

NonAssociativeRing

LeftModule(%)

CommutativeStar

Module(%)

SetCategory

LeftOreRing

CoercibleTo(OutputForm)

Rng

CommutativeRing

TwoSidedRecip

Magma

SemiGroup

BiModule(%, %)

AbelianGroup

AbelianSemiGroup

NonAssociativeSemiRing

NonAssociativeAlgebra(%)

NonAssociativeRng

unitsKnown

Ring

SemiRng

EntireRing

BasicType

SemiRing