Related Structures
Parent(R) : Rng -> Pow
Category(R) : Rng -> Cat
PrimeField(F) : Fld -> Fld
PrimeRing(R) : Rng -> Rng
Centre(R) : Rng -> Rng
Numerical Invariants
Characteristic(R) : Rng -> RngIntElt
# R : Rng -> RngIntElt
Predicates and Boolean Operations
IsCommutative(R) : Rng -> BoolElt
IsUnitary(R) : Rng -> BoolElt
IsFinite(R) : Rng -> BoolElt
IsOrdered(R) : Rng -> BoolElt
IsField(R) : Rng -> BoolElt
IsDivisionRing(R) : Rng -> BoolElt
IsEuclideanDomain(R) : Rng -> BoolElt
IsEuclideanRing(R) : Rng -> BoolElt
IsMagmaEuclideanRing(R) : Rng -> BoolElt
IsPID(R) : Rng -> BoolElt
IsPIR(R) : Rng -> BoolElt
IsUFD(R) : Rng -> BoolElt
IsDomain(R) : Rng -> BoolElt
HasGCD(R) : Rng -> BoolElt
R eq S : Rng, Rng -> Rng
R ne S : Rng, Rng -> Rng
Parent and Category
Parent(r) : RngElt -> Rng
Category(r) : RngElt -> Cat
Creation of Elements
Zero(R) : Rng -> RngElt
One(R) : Rng -> RngElt
R ! a : Rng, RngElt -> RngElt
Random(R) : Rng -> RngElt
Representative(R) : Rng -> RngElt
Arithmetic Operations
+ a : RngElt -> RngElt
- a : RngElt -> RngElt
a + b : RngElt, RngElt -> RngElt
a - b : RngElt, RngElt -> RngElt
a * b : RngElt, RngElt -> RngElt
a ^ k : RngElt, RngIntElt -> RngElt
a ^ -k : RngElt, RngIntElt -> RngElt
a / b : RngElt, RngElt -> RngElt
a +:= b : RngElt, RngElt -> RngElt
a -:= b : RngElt, RngElt -> RngElt
a *:= b : RngElt, RngElt -> RngElt
a /:= b : RngElt, RngElt -> RngElt
a ^:= k : RngElt, RngIntElt -> RngElt
Equality and Membership
a eq b : RngElt, RngElt -> BoolElt
a ne b : RngElt, RngElt -> BoolElt
R eq S : Rng, Rng -> BoolElt
R ne S : Rng, Rng -> BoolElt
a in R : RngElt, Rng -> BoolElt
a notin R : RngElt, Rng -> BoolElt
Predicates on Ring Elements
IsZero(a) : RngElt -> BoolElt
IsOne(a) : RngElt -> BoolElt
IsMinusOne(a) : RngElt -> BoolElt
IsUnit(a) : RngElt -> BoolElt
IsIdempotent(x) : RngElt -> BoolElt
IsNilpotent(x) : RngElt -> BoolElt
IsZeroDivisor(x) : RngElt -> BoolElt
IsIrreducible(x) : RngElt -> BoolElt
IsPrime(x) : RngElt -> BoolElt
Comparison of Ring Elements
a gt b : RngElt, RngElt -> BoolElt
a ge b : RngElt, RngElt -> BoolElt
a lt b : RngElt, RngElt -> BoolElt
a le b : RngElt, RngElt -> BoolElt
Maximum(a, b) : RngElt, RngElt -> RngElt
Maximum(Q) : [RngIntElt] -> RngElt
Minimum(a, b) : RngElt, RngElt -> RngElt
Minimum(Q) : [RngIntElt] -> RngElt
Defining Ideals and Quotient Rings
ideal< R | a1, ..., ar > : Rng, RngElt, ..., RngElt -> RngIdl
quo< R | ar, ..., ar > : Rng, RngElt, ..., RngElt -> Rng
R / I : Rng, RngIdl -> Rng
PowerIdeal(R) : Rng -> PowIdl
Arithmetic Operations on Ideals
I + J : RngIdl, RngIdl -> RngIdl
I * J : RngIdl, RngIdl -> RngIdl
I meet J : RngIdl, RngIdl -> RngIdl
Boolean Operators on Ideals
a in I : RngElt, RngIdl -> BoolElt
a notin I : RngElt, RngIdl -> BoolElt
I eq J : RngIdl, RngIdl -> BoolElt
I ne J : RngIdl, RngIdl -> BoolElt
I subset J : RngIdl, RngIdl -> BoolElt
I notsubset J : RngIdl, RngIdl -> BoolElt
Residue Class Fields
ResidueClassField(I) : Rng -> Fld, Map
Localization
loc< R | a1, ..., ar > : Rng, RngElt, ..., RngElt -> Rng, Map
Localization(R, P) : Rng, Rng -> Rng, Map
Completion
comp< R | a1, ..., ar > : Rng, RngElt, ..., RngElt -> Rng, Map
Completion(R, P) : Rng, Rng -> Rng, Map
Transcendental Extension
ext< R | > : Rng -> RngUPol
ext< R, n | > : Rng, RngIntElt -> RngMPol
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Wed Apr 24 15:09:57 EST 2013