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INTRODUCTION TO RINGS

 
Acknowledgements
 
Overview
 
The World of Rings
      New Rings from Existing Ones
      Attributes
 
Coercion
      Automatic Coercion
      Forced Coercion
 
Generic Ring Functions
      Related Structures
      Numerical Invariants
      Predicates and Boolean Operations
 
Generic Element Functions
      Parent and Category
      Creation of Elements
      Arithmetic Operations
      Equality and Membership
      Predicates on Ring Elements
      Comparison of Ring Elements
 
Ideals and Quotient Rings
      Defining Ideals and Quotient Rings
      Arithmetic Operations on Ideals
      Boolean Operators on Ideals
 
Other Ring Constructions
      Residue Class Fields
      Localization
      Completion
      Transcendental Extension







DETAILS

 
Overview

 
The World of Rings

      New Rings from Existing Ones

      Attributes

 
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 ^ -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, ..., 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

 
Other Ring Constructions

      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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Version: V2.19 of Wed Apr 24 15:09:57 EST 2013