Introduction to Rings

• 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
    • Type(R): Rng → Cat
    • PrimeField(F): Fld → Fld
    • PrimeRing(R): Rng → Rng
    • Centre(R): Rng → Rng
    • Center(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
    • IsPrincipalIdealDomain(R): Rng → BoolElt
    • IsPIR(R): Rng → BoolElt
    • IsPrincipalIdealRing(R): Rng → BoolElt
    • IsUFD(R): Rng → BoolElt
    • IsUniqueFactorizationDomain(R): Rng → BoolElt
    • IsDomain(R): Rng → BoolElt
    • IsIntegralDomain(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
    • Type(r): RngElt → Cat
  • Creation of Elements
    • Zero(R): Rng → RngElt
    • One(R): Rng → RngElt
    • Id(R): Rng → RngElt
    • R ! a: Rng, RngElt → RngElt
    • Random(R): Rng → RngElt
    • Representative(R): Rng → RngElt
    • Rep(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, ..., 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