EntireRingΒΆ

catdef.spad line 349

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 Magma
*: (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: (%, %) -> %
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: % -> OutputForm
from CoercibleTo OutputForm
coerce: Integer -> %
from NonAssociativeRing
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.
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)
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

AbelianGroup

AbelianMonoid

AbelianSemiGroup

BasicType

BiModule(%, %)

CancellationAbelianMonoid

CoercibleTo OutputForm

LeftModule %

Magma

MagmaWithUnit

Monoid

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors

RightModule %

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

unitsKnown