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 #
RingCon R
: the type of congruence relations respecting+
and*
.RingConGen r
: the inductively defined smallest ring congruence relation containing a given binary relation.
TODO #
- Use this for
RingQuot
too. - Copy across more API from
Con
andAddCon
inGroupTheory/Congruence.lean
, such as:- The
CompleteLattice
structure. - The
conGen_eq
lemma, stating thatringConGen r = sInf {s : RingCon M | ∀ x y, r x y → s x y}
.
- The
- r : R → R → Prop
- iseqv : Equivalence Setoid.r
Additive congruence relations are closed under addition
A congruence relation on a type with an addition and multiplication is an equivalence relation which preserves both.
Instances For
- of: ∀ {R : Type u_2} [inst : Add R] [inst_1 : Mul R] {r : R → R → Prop} (x y : R), r x y → RingConGen.Rel r x y
- refl: ∀ {R : Type u_2} [inst : Add R] [inst_1 : Mul R] {r : R → R → Prop} (x : R), RingConGen.Rel r x x
- symm: ∀ {R : Type u_2} [inst : Add R] [inst_1 : Mul R] {r : R → R → Prop} {x y : R}, RingConGen.Rel r x y → RingConGen.Rel r y x
- trans: ∀ {R : Type u_2} [inst : Add R] [inst_1 : Mul R] {r : R → R → Prop} {x y z : R}, RingConGen.Rel r x y → RingConGen.Rel r y z → RingConGen.Rel r x z
- add: ∀ {R : Type u_2} [inst : Add R] [inst_1 : Mul R] {r : R → R → Prop} {w x y z : R}, RingConGen.Rel r w x → RingConGen.Rel r y z → RingConGen.Rel r (w + y) (x + z)
- mul: ∀ {R : Type u_2} [inst : Add R] [inst_1 : Mul R] {r : R → R → Prop} {w x y z : R}, RingConGen.Rel r w x → RingConGen.Rel r y z → RingConGen.Rel r (w * y) (x * z)
The inductively defined smallest ring congruence relation containing a given binary relation.
Instances For
The morphism into the quotient by a congruence relation
Instances For
Coercion from a type with addition and multiplication to its quotient by a congruence relation.
See Note [use has_coe_t].
Basic notation #
The basic algebraic notation, 0
, 1
, +
, *
, -
, ^
, descend naturally under the quotient
Algebraic structure #
The operations above on the quotient by c : RingCon R
preserve the algebraic structure of R
.
The natural homomorphism from a ring to its quotient by a congruence relation.