Overview

 
The World of Rings

      New Rings from Existing Ones

 
Coercion

      Automatic Coercion

      Forced Coercion

 
Generic Ring Functions

      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

 
Generic Element Functions

      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 / 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

 
Ideals and Quotient Rings

      Defining Ideals and Quotient Rings
            ideal< R | a₁, ..., a_r > : Rng, RngElt, ..., RngElt -> RngIdl
            quo< R | a_r, ..., a_r > : 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

 
Other Ring Constructions

      Residue Class Fields
            ResidueClassField(I) : Rng -> Fld, Map

      Localization
            loc< R | a₁, ..., a_r > : Rng, RngElt, ..., RngElt -> Rng, Map
            Localization(R, P) : Rng, Rng -> Rng, Map

      Completion
            comp< R | a₁, ..., a_r > : 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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