# FiniteAbelianMonoidRing(R, E)¶

This category is similar to AbelianMonoidRing, except that the sum is assumed to be finite. It is a useful model for polynomials, but is somewhat more general.

0: %

from AbelianMonoid

1: % if R has SemiRing

from MagmaWithUnit

*: (%, %) -> %

from LeftModule %

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

from RightModule R

*: (Fraction Integer, %) -> % if R has Algebra Fraction Integer
*: (Integer, %) -> % if R has AbelianGroup or % has AbelianGroup

from AbelianGroup

*: (NonNegativeInteger, %) -> %

from AbelianMonoid

*: (PositiveInteger, %) -> %

from AbelianSemiGroup

*: (R, %) -> %

from LeftModule R

+: (%, %) -> %

from AbelianSemiGroup

-: % -> % if R has AbelianGroup or % has AbelianGroup

from AbelianGroup

-: (%, %) -> % if R has AbelianGroup or % has AbelianGroup

from AbelianGroup

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

from AbelianMonoidRing(R, E)

=: (%, %) -> Boolean

from BasicType

^: (%, NonNegativeInteger) -> % if R has SemiRing

from MagmaWithUnit

^: (%, PositiveInteger) -> %

from Magma

~=: (%, %) -> Boolean

from BasicType

annihilate?: (%, %) -> Boolean if R has Ring

from Rng

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

from EntireRing

associator: (%, %, %) -> % if R has Ring
binomThmExpt: (%, %, NonNegativeInteger) -> % if % has CommutativeRing

binomThmExpt(p, q, n) returns (p+q)^n by means of the binomial theorem trick.

characteristic: () -> NonNegativeInteger if R has Ring
charthRoot: % -> Union(%, failed) if R has CharacteristicNonZero
coefficient: (%, E) -> R

from AbelianMonoidRing(R, E)

coefficients: % -> List R

from FreeModuleCategory(R, E)

coerce: % -> % if R has IntegralDomain and % has VariablesCommuteWithCoefficients or R has CommutativeRing and % has VariablesCommuteWithCoefficients

from Algebra %

coerce: % -> OutputForm
coerce: Fraction Integer -> % if R has RetractableTo Fraction Integer or R has Algebra Fraction Integer
coerce: Integer -> % if R has Ring or R has RetractableTo Integer
coerce: R -> %

from Algebra R

commutator: (%, %) -> % if R has Ring
construct: List Record(k: E, c: R) -> %

from IndexedProductCategory(R, E)

constructOrdered: List Record(k: E, c: R) -> %

from IndexedProductCategory(R, E)

content: % -> R if R has GcdDomain

content(p) gives the gcd of the coefficients of polynomial p.

degree: % -> E

from AbelianMonoidRing(R, E)

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

from EntireRing

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

exquo(p,r) returns the exact quotient of polynomial p by r, or “failed” if none exists.

fmecg: (%, E, R, %) -> % if R has Ring

fmecg(p1, e, r, p2) returns p1 - monomial(r, e) * p2.

ground?: % -> Boolean

ground?(p) tests if polynomial p is a member of the coefficient ring.

ground: % -> R

ground(p) retracts polynomial p to the coefficient ring.

hash: % -> SingleInteger

from SetCategory

hashUpdate!: (HashState, %) -> HashState

from SetCategory

latex: % -> String

from SetCategory

from IndexedProductCategory(R, E)

from IndexedProductCategory(R, E)

from IndexedProductCategory(R, E)

leadingTerm: % -> Record(k: E, c: R)

from IndexedProductCategory(R, E)

leftPower: (%, NonNegativeInteger) -> % if R has SemiRing

from MagmaWithUnit

leftPower: (%, PositiveInteger) -> %

from Magma

leftRecip: % -> Union(%, failed) if R has SemiRing

from MagmaWithUnit

linearExtend: (E -> R, %) -> R if R has CommutativeRing

from FreeModuleCategory(R, E)

listOfTerms: % -> List Record(k: E, c: R)

from IndexedDirectProductCategory(R, E)

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

from IndexedProductCategory(R, E)

mapExponents: (E -> E, %) -> %

mapExponents(fn, u) maps function fn onto the exponents of the non-zero monomials of polynomial u.

minimumDegree: % -> E

minimumDegree(p) gives the least exponent of a non-zero term of polynomial p. Error: if applied to 0.

monomial?: % -> Boolean

from IndexedProductCategory(R, E)

monomial: (R, E) -> %

from IndexedProductCategory(R, E)

monomials: % -> List %

from FreeModuleCategory(R, E)

numberOfMonomials: % -> NonNegativeInteger

from IndexedDirectProductCategory(R, E)

one?: % -> Boolean if R has SemiRing

from MagmaWithUnit

opposite?: (%, %) -> Boolean

from AbelianMonoid

pomopo!: (%, R, E, %) -> %

pomopo!(p1, r, e, p2) returns p1 + monomial(r, e) * p2 and may use p1 as workspace. The constant r is assumed to be nonzero.

primitivePart: % -> % if R has GcdDomain

primitivePart(p) returns the unit normalized form of polynomial p divided by the content of p.

recip: % -> Union(%, failed) if R has SemiRing

from MagmaWithUnit

reductum: % -> %

from IndexedProductCategory(R, E)

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

from RetractableTo R

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

from RetractableTo R

rightPower: (%, NonNegativeInteger) -> % if R has SemiRing

from MagmaWithUnit

rightPower: (%, PositiveInteger) -> %

from Magma

rightRecip: % -> Union(%, failed) if R has SemiRing

from MagmaWithUnit

sample: %

from AbelianMonoid

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

from Comparable

subtractIfCan: (%, %) -> Union(%, failed)
support: % -> List E

from FreeModuleCategory(R, E)

unit?: % -> Boolean if R has EntireRing

from EntireRing

unitCanonical: % -> % if R has EntireRing

from EntireRing

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

from EntireRing

zero?: % -> Boolean

from AbelianMonoid

AbelianGroup if R has AbelianGroup

AbelianMonoid

AbelianMonoidRing(R, E)

AbelianSemiGroup

Algebra % if R has IntegralDomain and % has VariablesCommuteWithCoefficients or R has CommutativeRing and % has VariablesCommuteWithCoefficients

Algebra R if R has CommutativeRing and % has VariablesCommuteWithCoefficients

BasicType

BiModule(%, %)

BiModule(R, R)

CancellationAbelianMonoid

CommutativeRing if R has CommutativeRing and % has VariablesCommuteWithCoefficients or R has IntegralDomain and % has VariablesCommuteWithCoefficients

CommutativeStar if R has IntegralDomain and % has VariablesCommuteWithCoefficients or R has CommutativeRing and % has VariablesCommuteWithCoefficients

Comparable if R has Comparable

EntireRing if R has EntireRing

FreeModuleCategory(R, E)

IndexedProductCategory(R, E)

IntegralDomain if R has IntegralDomain and % has VariablesCommuteWithCoefficients

Magma

MagmaWithUnit if R has SemiRing

Module % if R has IntegralDomain and % has VariablesCommuteWithCoefficients or R has CommutativeRing and % has VariablesCommuteWithCoefficients

Module R if R has CommutativeRing

Monoid if R has SemiRing

NonAssociativeRing if R has Ring

NonAssociativeRng if R has Ring

NonAssociativeSemiRing if R has SemiRing

NonAssociativeSemiRng

noZeroDivisors if R has EntireRing

Ring if R has Ring

Rng if R has Ring

SemiGroup

SemiRing if R has SemiRing

SemiRng

SetCategory

TwoSidedRecip if R has IntegralDomain and % has VariablesCommuteWithCoefficients or R has CommutativeRing and % has VariablesCommuteWithCoefficients

unitsKnown if R has Ring