# EntireRing¶

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

0: %

from AbelianMonoid

1: %

from MagmaWithUnit

*: (%, %) -> %

from LeftModule %

*: (Integer, %) -> %

from AbelianGroup

*: (NonNegativeInteger, %) -> %

from AbelianMonoid

*: (PositiveInteger, %) -> %

from AbelianSemiGroup

+: (%, %) -> %

from AbelianSemiGroup

-: % -> %

from AbelianGroup

-: (%, %) -> %

from AbelianGroup

=: (%, %) -> Boolean

from BasicType

^: (%, NonNegativeInteger) -> %

from MagmaWithUnit

^: (%, PositiveInteger) -> %

from Magma

~=: (%, %) -> Boolean

from BasicType

annihilate?: (%, %) -> Boolean

from Rng

antiCommutator: (%, %) -> %
associates?: (%, %) -> Boolean

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

associator: (%, %, %) -> %
characteristic: () -> NonNegativeInteger
coerce: % -> OutputForm
coerce: Integer -> %
commutator: (%, %) -> %
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.

hash: % -> SingleInteger

from SetCategory

hashUpdate!: (HashState, %) -> HashState

from SetCategory

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)
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

AbelianGroup

AbelianMonoid

AbelianSemiGroup

BasicType

BiModule(%, %)

CancellationAbelianMonoid

Magma

MagmaWithUnit

Monoid

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

unitsKnown