Documentation

Mathlib.RingTheory.Congruence

Congruence relations on rings #

This file defines congruence relations on rings, which extend Con and AddCon on monoids and additive monoids.

Most of the time you likely want to use the Ideal.Quotient API that is built on top of this.

Main Definitions #

TODO #

structure RingCon (R : Type u_1) [Add R] [Mul R] extends Con :
Type u_1

A congruence relation on a type with an addition and multiplication is an equivalence relation which preserves both.

Instances For
    @[reducible]
    abbrev RingCon.toAddCon {R : Type u_1} [Add R] [Mul R] (self : RingCon R) :

    The induced additive congruence from a RingCon.

    Equations
    • self.toAddCon = { toSetoid := self.toSetoid, add' := }
    Instances For
      inductive RingConGen.Rel {R : Type u_2} [Add R] [Mul R] (r : RRProp) :
      RRProp

      The inductively defined smallest ring congruence relation containing a given binary relation.

      Instances For
        def ringConGen {R : Type u_2} [Add R] [Mul R] (r : RRProp) :

        The inductively defined smallest ring congruence relation containing a given binary relation.

        Equations
        Instances For
          instance RingCon.instFunLikeForallProp {R : Type u_2} [Add R] [Mul R] :
          FunLike (RingCon R) R (RProp)

          A coercion from a congruence relation to its underlying binary relation.

          Equations
          • RingCon.instFunLikeForallProp = { coe := fun (c : RingCon R) => Setoid.r, coe_injective' := }
          theorem RingCon.rel_eq_coe {R : Type u_2} [Add R] [Mul R] (c : RingCon R) :
          Setoid.r = c
          @[simp]
          theorem RingCon.toCon_coe_eq_coe {R : Type u_2} [Add R] [Mul R] (c : RingCon R) :
          c.toCon = c
          theorem RingCon.refl {R : Type u_2} [Add R] [Mul R] (c : RingCon R) (x : R) :
          c x x
          theorem RingCon.symm {R : Type u_2} [Add R] [Mul R] (c : RingCon R) {x : R} {y : R} :
          c x yc y x
          theorem RingCon.trans {R : Type u_2} [Add R] [Mul R] (c : RingCon R) {x : R} {y : R} {z : R} :
          c x yc y zc x z
          theorem RingCon.add {R : Type u_2} [Add R] [Mul R] (c : RingCon R) {w : R} {x : R} {y : R} {z : R} :
          c w xc y zc (w + y) (x + z)
          theorem RingCon.mul {R : Type u_2} [Add R] [Mul R] (c : RingCon R) {w : R} {x : R} {y : R} {z : R} :
          c w xc y zc (w * y) (x * z)
          instance RingCon.instInhabited {R : Type u_2} [Add R] [Mul R] :
          Equations
          • RingCon.instInhabited = { default := ringConGen EmptyRelation }
          @[simp]
          theorem RingCon.rel_mk {R : Type u_2} [Add R] [Mul R] {s : Con R} {h : ∀ {w x y z : R}, Setoid.r w xSetoid.r y zSetoid.r (w + y) (x + z)} {a : R} {b : R} :
          { toCon := s, add' := h } a b s a b
          theorem RingCon.ext' {R : Type u_2} [Add R] [Mul R] {c : RingCon R} {d : RingCon R} (H : c = d) :
          c = d

          The map sending a congruence relation to its underlying binary relation is injective.

          theorem RingCon.ext {R : Type u_2} [Add R] [Mul R] {c : RingCon R} {d : RingCon R} (H : ∀ (x y : R), c x y d x y) :
          c = d

          Extensionality rule for congruence relations.

          def RingCon.Quotient {R : Type u_2} [Add R] [Mul R] (c : RingCon R) :
          Type u_2

          Defining the quotient by a congruence relation of a type with addition and multiplication.

          Equations
          Instances For
            def RingCon.toQuotient {R : Type u_2} [Add R] [Mul R] {c : RingCon R} (r : R) :
            c.Quotient

            The morphism into the quotient by a congruence relation

            Equations
            Instances For
              instance RingCon.instCoeTCQuotient {R : Type u_2} [Add R] [Mul R] (c : RingCon R) :
              CoeTC R c.Quotient

              Coercion from a type with addition and multiplication to its quotient by a congruence relation.

              See Note [use has_coe_t].

              Equations
              • c.instCoeTCQuotient = { coe := RingCon.toQuotient }
              @[instance 500]
              instance RingCon.instDecidableEqQuotientOfDecidableCoeForallProp {R : Type u_2} [Add R] [Mul R] (c : RingCon R) [_d : (a b : R) → Decidable (c a b)] :
              DecidableEq c.Quotient

              The quotient by a decidable congruence relation has decidable equality.

              Equations
              @[simp]
              theorem RingCon.quot_mk_eq_coe {R : Type u_2} [Add R] [Mul R] (c : RingCon R) (x : R) :
              Quot.mk (c) x = x
              @[simp]
              theorem RingCon.eq {R : Type u_2} [Add R] [Mul R] (c : RingCon R) {a : R} {b : R} :
              a = b c a b

              Two elements are related by a congruence relation c iff they are represented by the same element of the quotient by c.

              Basic notation #

              The basic algebraic notation, 0, 1, +, *, -, ^, descend naturally under the quotient

              instance RingCon.instAddQuotient {R : Type u_2} [Add R] [Mul R] (c : RingCon R) :
              Add c.Quotient
              Equations
              @[simp]
              theorem RingCon.coe_add {R : Type u_2} [Add R] [Mul R] (c : RingCon R) (x : R) (y : R) :
              (x + y) = x + y
              instance RingCon.instMulQuotient {R : Type u_2} [Add R] [Mul R] (c : RingCon R) :
              Mul c.Quotient
              Equations
              @[simp]
              theorem RingCon.coe_mul {R : Type u_2} [Add R] [Mul R] (c : RingCon R) (x : R) (y : R) :
              (x * y) = x * y
              instance RingCon.instZeroQuotient {R : Type u_2} [AddZeroClass R] [Mul R] (c : RingCon R) :
              Zero c.Quotient
              Equations
              @[simp]
              theorem RingCon.coe_zero {R : Type u_2} [AddZeroClass R] [Mul R] (c : RingCon R) :
              0 = 0
              instance RingCon.instOneQuotient {R : Type u_2} [Add R] [MulOneClass R] (c : RingCon R) :
              One c.Quotient
              Equations
              @[simp]
              theorem RingCon.coe_one {R : Type u_2} [Add R] [MulOneClass R] (c : RingCon R) :
              1 = 1
              instance RingCon.instSMulQuotient {α : Type u_1} {R : Type u_2} [Add R] [MulOneClass R] [SMul α R] [IsScalarTower α R R] (c : RingCon R) :
              SMul α c.Quotient
              Equations
              @[simp]
              theorem RingCon.coe_smul {α : Type u_1} {R : Type u_2} [Add R] [MulOneClass R] [SMul α R] [IsScalarTower α R R] (c : RingCon R) (a : α) (x : R) :
              (a x) = a x
              instance RingCon.instNegQuotient {R : Type u_2} [AddGroup R] [Mul R] (c : RingCon R) :
              Neg c.Quotient
              Equations
              @[simp]
              theorem RingCon.coe_neg {R : Type u_2} [AddGroup R] [Mul R] (c : RingCon R) (x : R) :
              (-x) = -x
              instance RingCon.instSubQuotient {R : Type u_2} [AddGroup R] [Mul R] (c : RingCon R) :
              Sub c.Quotient
              Equations
              @[simp]
              theorem RingCon.coe_sub {R : Type u_2} [AddGroup R] [Mul R] (c : RingCon R) (x : R) (y : R) :
              (x - y) = x - y
              instance RingCon.hasZSMul {R : Type u_2} [AddGroup R] [Mul R] (c : RingCon R) :
              SMul c.Quotient
              Equations
              @[simp]
              theorem RingCon.coe_zsmul {R : Type u_2} [AddGroup R] [Mul R] (c : RingCon R) (z : ) (x : R) :
              (z x) = z x
              instance RingCon.hasNSMul {R : Type u_2} [AddMonoid R] [Mul R] (c : RingCon R) :
              SMul c.Quotient
              Equations
              @[simp]
              theorem RingCon.coe_nsmul {R : Type u_2} [AddMonoid R] [Mul R] (c : RingCon R) (n : ) (x : R) :
              (n x) = n x
              instance RingCon.instPowQuotientNat {R : Type u_2} [Add R] [Monoid R] (c : RingCon R) :
              Pow c.Quotient
              Equations
              @[simp]
              theorem RingCon.coe_pow {R : Type u_2} [Add R] [Monoid R] (c : RingCon R) (x : R) (n : ) :
              (x ^ n) = x ^ n
              instance RingCon.instNatCastQuotient {R : Type u_2} [AddMonoidWithOne R] [Mul R] (c : RingCon R) :
              NatCast c.Quotient
              Equations
              • c.instNatCastQuotient = { natCast := fun (n : ) => n }
              @[simp]
              theorem RingCon.coe_natCast {R : Type u_2} [AddMonoidWithOne R] [Mul R] (c : RingCon R) (n : ) :
              n = n
              @[deprecated RingCon.coe_natCast]
              theorem RingCon.coe_nat_cast {R : Type u_2} [AddMonoidWithOne R] [Mul R] (c : RingCon R) (n : ) :
              n = n

              Alias of RingCon.coe_natCast.

              instance RingCon.instIntCastQuotient {R : Type u_2} [AddGroupWithOne R] [Mul R] (c : RingCon R) :
              IntCast c.Quotient
              Equations
              • c.instIntCastQuotient = { intCast := fun (z : ) => z }
              @[simp]
              theorem RingCon.coe_intCast {R : Type u_2} [AddGroupWithOne R] [Mul R] (c : RingCon R) (n : ) :
              n = n
              @[deprecated RingCon.coe_intCast]
              theorem RingCon.coe_int_cast {R : Type u_2} [AddGroupWithOne R] [Mul R] (c : RingCon R) (n : ) :
              n = n

              Alias of RingCon.coe_intCast.

              instance RingCon.instInhabitedQuotient {R : Type u_2} [Inhabited R] [Add R] [Mul R] (c : RingCon R) :
              Inhabited c.Quotient
              Equations
              • c.instInhabitedQuotient = { default := default }

              Algebraic structure #

              The operations above on the quotient by c : RingCon R preserve the algebraic structure of R.

              Equations
              Equations
              Equations
              instance RingCon.instSemiringQuotient {R : Type u_2} [Semiring R] (c : RingCon R) :
              Semiring c.Quotient
              Equations
              instance RingCon.instCommSemiringQuotient {R : Type u_2} [CommSemiring R] (c : RingCon R) :
              CommSemiring c.Quotient
              Equations
              Equations
              instance RingCon.instNonAssocRingQuotient {R : Type u_2} [NonAssocRing R] (c : RingCon R) :
              NonAssocRing c.Quotient
              Equations
              instance RingCon.instNonUnitalRingQuotient {R : Type u_2} [NonUnitalRing R] (c : RingCon R) :
              NonUnitalRing c.Quotient
              Equations
              instance RingCon.instRingQuotient {R : Type u_2} [Ring R] (c : RingCon R) :
              Ring c.Quotient
              Equations
              instance RingCon.instCommRingQuotient {R : Type u_2} [CommRing R] (c : RingCon R) :
              CommRing c.Quotient
              Equations
              instance RingCon.isScalarTower_right {α : Type u_1} {R : Type u_2} [Add R] [MulOneClass R] [SMul α R] [IsScalarTower α R R] (c : RingCon R) :
              IsScalarTower α c.Quotient c.Quotient
              Equations
              • =
              instance RingCon.smulCommClass {α : Type u_1} {R : Type u_2} [Add R] [MulOneClass R] [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] (c : RingCon R) :
              SMulCommClass α c.Quotient c.Quotient
              Equations
              • =
              instance RingCon.smulCommClass' {α : Type u_1} {R : Type u_2} [Add R] [MulOneClass R] [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] (c : RingCon R) :
              SMulCommClass c.Quotient α c.Quotient
              Equations
              • =
              Equations
              • c.instDistribMulActionQuotientOfIsScalarTower = let __src := c.mulAction; DistribMulAction.mk
              instance RingCon.instMulSemiringActionQuotientOfIsScalarTower {α : Type u_1} {R : Type u_2} [Monoid α] [Semiring R] [MulSemiringAction α R] [IsScalarTower α R R] (c : RingCon R) :
              MulSemiringAction α c.Quotient
              Equations
              def RingCon.mk' {R : Type u_2} [NonAssocSemiring R] (c : RingCon R) :
              R →+* c.Quotient

              The natural homomorphism from a ring to its quotient by a congruence relation.

              Equations
              • c.mk' = { toFun := RingCon.toQuotient, map_one' := , map_mul' := , map_zero' := , map_add' := }
              Instances For

                Lattice structure #

                The API in this section is copied from Mathlib/GroupTheory/Congruence.lean

                instance RingCon.instLE {R : Type u_2} [Add R] [Mul R] :

                For congruence relations c, d on a type M with multiplication and addition, c ≤ d iff ∀ x y ∈ M, x is related to y by d if x is related to y by c.

                Equations
                • RingCon.instLE = { le := fun (c d : RingCon R) => ∀ ⦃x y : R⦄, c x yd x y }
                theorem RingCon.le_def {R : Type u_2} [Add R] [Mul R] {c : RingCon R} {d : RingCon R} :
                c d ∀ {x y : R}, c x yd x y

                Definition of for congruence relations.

                instance RingCon.instInfSet {R : Type u_2} [Add R] [Mul R] :

                The infimum of a set of congruence relations on a given type with multiplication and addition.

                Equations
                • RingCon.instInfSet = { sInf := fun (S : Set (RingCon R)) => { r := fun (x y : R) => cS, c x y, iseqv := , mul' := , add' := } }
                theorem RingCon.sInf_toSetoid {R : Type u_2} [Add R] [Mul R] (S : Set (RingCon R)) :
                (sInf S).toSetoid = sInf ((fun (x : RingCon R) => x.toSetoid) '' S)

                The infimum of a set of congruence relations is the same as the infimum of the set's image under the map to the underlying equivalence relation.

                @[simp]
                theorem RingCon.coe_sInf {R : Type u_2} [Add R] [Mul R] (S : Set (RingCon R)) :
                (sInf S) = sInf (DFunLike.coe '' S)

                The infimum of a set of congruence relations is the same as the infimum of the set's image under the map to the underlying binary relation.

                @[simp]
                theorem RingCon.coe_iInf {R : Type u_2} [Add R] [Mul R] {ι : Sort u_3} (f : ιRingCon R) :
                (iInf f) = ⨅ (i : ι), (f i)
                instance RingCon.instPartialOrder {R : Type u_2} [Add R] [Mul R] :
                Equations

                The complete lattice of congruence relations on a given type with multiplication and addition.

                Equations
                @[simp]
                theorem RingCon.coe_top {R : Type u_2} [Add R] [Mul R] :
                =
                @[simp]
                theorem RingCon.coe_bot {R : Type u_2} [Add R] [Mul R] :
                = Eq
                @[simp]
                theorem RingCon.coe_inf {R : Type u_2} [Add R] [Mul R] {c : RingCon R} {d : RingCon R} :
                (c d) = c d

                The infimum of two congruence relations equals the infimum of the underlying binary operations.

                theorem RingCon.inf_iff_and {R : Type u_2} [Add R] [Mul R] {c : RingCon R} {d : RingCon R} {x : R} {y : R} :
                (c d) x y c x y d x y

                Definition of the infimum of two congruence relations.

                instance RingCon.instNontrivial {R : Type u_2} [Add R] [Mul R] [Nontrivial R] :
                Equations
                • =
                theorem RingCon.ringConGen_eq {R : Type u_2} [Add R] [Mul R] (r : RRProp) :
                ringConGen r = sInf {s : RingCon R | ∀ (x y : R), r x ys x y}

                The inductively defined smallest congruence relation containing a binary relation r equals the infimum of the set of congruence relations containing r.

                theorem RingCon.ringConGen_le {R : Type u_2} [Add R] [Mul R] {r : RRProp} {c : RingCon R} (h : ∀ (x y : R), r x yc x y) :

                The smallest congruence relation containing a binary relation r is contained in any congruence relation containing r.

                theorem RingCon.ringConGen_mono {R : Type u_2} [Add R] [Mul R] {r : RRProp} {s : RRProp} (h : ∀ (x y : R), r x ys x y) :

                Given binary relations r, s with r contained in s, the smallest congruence relation containing s contains the smallest congruence relation containing r.

                theorem RingCon.ringConGen_of_ringCon {R : Type u_2} [Add R] [Mul R] (c : RingCon R) :
                ringConGen c = c

                Congruence relations equal the smallest congruence relation in which they are contained.

                theorem RingCon.ringConGen_idem {R : Type u_2} [Add R] [Mul R] (r : RRProp) :

                The map sending a binary relation to the smallest congruence relation in which it is contained is idempotent.

                theorem RingCon.sup_eq_ringConGen {R : Type u_2} [Add R] [Mul R] (c : RingCon R) (d : RingCon R) :
                c d = ringConGen fun (x y : R) => c x y d x y

                The supremum of congruence relations c, d equals the smallest congruence relation containing the binary relation 'x is related to y by c or d'.

                theorem RingCon.sup_def {R : Type u_2} [Add R] [Mul R] {c : RingCon R} {d : RingCon R} :
                c d = ringConGen (c d)

                The supremum of two congruence relations equals the smallest congruence relation containing the supremum of the underlying binary operations.

                theorem RingCon.sSup_eq_ringConGen {R : Type u_2} [Add R] [Mul R] (S : Set (RingCon R)) :
                sSup S = ringConGen fun (x y : R) => cS, c x y

                The supremum of a set of congruence relations S equals the smallest congruence relation containing the binary relation 'there exists c ∈ S such that x is related to y by c'.

                theorem RingCon.sSup_def {R : Type u_2} [Add R] [Mul R] {S : Set (RingCon R)} :
                sSup S = ringConGen (sSup (DFunLike.coe '' S))

                The supremum of a set of congruence relations is the same as the smallest congruence relation containing the supremum of the set's image under the map to the underlying binary relation.

                def RingCon.gi (R : Type u_2) [Add R] [Mul R] :
                GaloisInsertion ringConGen DFunLike.coe

                There is a Galois insertion of congruence relations on a type with multiplication and addition R into binary relations on R.

                Equations
                Instances For