EntireRing

catdef.spad line 350 [edit on github]

Entire Rings (non-commutative Integral Domains), i.e. a ring not necessarily commutative which has no zero divisors.

* : (%, %) -> %

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

associates?(x, y) tests whether x and y are associates, i.e. differ by a unit factor.

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

from NonAssociativeRng

characteristic : () -> NonNegativeInteger

from NonAssociativeRing

coerce : Integer -> %

from NonAssociativeRing

coerce : % -> OutputForm

from CoercibleTo(OutputForm)

commutator : (%, %) -> %

from NonAssociativeRng

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

exquo(a, b) either returns an element c such that c*b=a or "failed" if no such element can be found.

latex : % -> String

from SetCategory

leftPower : (%, NonNegativeInteger) -> %

from MagmaWithUnit

leftPower : (%, PositiveInteger) -> %

from Magma

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

from MagmaWithUnit

one? : % -> Boolean

from MagmaWithUnit

opposite? : (%, %) -> Boolean

from AbelianMonoid

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

unit?(x) tests whether x is a unit, i.e. is invertible.

unitCanonical : % -> %

unitCanonical(x) returns unitNormal(x).canonical.

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

unitNormal(x) tries to choose a canonical element from the associate class of x. The attribute canonicalUnitNormal, if asserted, means that the "canonical" element is the same across all associates of x if unitNormal(x) = [u, c, a] then u*c = x, a*u = 1.

zero? : % -> Boolean

from AbelianMonoid

~= : (%, %) -> Boolean

from BasicType

RightModule(%)

Rng

Monoid

noZeroDivisors

Ring

SemiGroup

CancellationAbelianMonoid

LeftModule(%)

BasicType

unitsKnown

NonAssociativeRing

Magma

NonAssociativeSemiRng

SemiRing

AbelianGroup

NonAssociativeSemiRing

SetCategory

AbelianSemiGroup

AbelianMonoid

BiModule(%, %)

MagmaWithUnit

CoercibleTo(OutputForm)

SemiRng

NonAssociativeRng