Documentation

Mathlib.Topology.Algebra.Ring.Basic

Topological (semi)rings #

A topological (semi)ring is a (semi)ring equipped with a topology such that all operations are continuous. Besides this definition, this file proves that the topological closure of a subring (resp. an ideal) is a subring (resp. an ideal) and defines products and quotients of topological (semi)rings.

Main Results #

    a topological semiring is a semiring R where addition and multiplication are continuous. We allow for non-unital and non-associative semirings as well.

    The TopologicalSemiring class should only be instantiated in the presence of a NonUnitalNonAssocSemiring instance; if there is an instance of NonUnitalNonAssocRing, then TopologicalRing should be used. Note: in the presence of NonAssocRing, these classes are mathematically equivalent (see TopologicalSemiring.continuousNeg_of_mul or TopologicalSemiring.toTopologicalRing).

    Instances

        A topological ring is a ring R where addition, multiplication and negation are continuous.

        If R is a (unital) ring, then continuity of negation can be derived from continuity of multiplication as it is multiplication with -1. (See TopologicalSemiring.continuousNeg_of_mul and topological_semiring.to_topological_add_group)

        Instances

          If R is a ring with a continuous multiplication, then negation is continuous as well since it is just multiplication with -1.

          If R is a ring which is a topological semiring, then it is automatically a topological ring. This exists so that one can place a topological ring structure on R without explicitly proving continuous_neg.

          The (topological-space) closure of a subsemiring of a topological semiring is itself a subsemiring.

          Instances For
            def Subsemiring.commSemiringTopologicalClosure {α : Type u_1} [TopologicalSpace α] [Semiring α] [TopologicalSemiring α] [T2Space α] (s : Subsemiring α) (hs : ∀ (x y : { x // x s }), x * y = y * x) :

            If a subsemiring of a topological semiring is commutative, then so is its topological closure.

            Instances For

              The product topology on the cartesian product of two topological semirings makes the product into a topological semiring.

              The product topology on the cartesian product of two topological rings makes the product into a topological ring.

              instance Pi.instTopologicalSemiring {β : Type u_2} {C : βType u_3} [(b : β) → TopologicalSpace (C b)] [(b : β) → NonUnitalNonAssocSemiring (C b)] [∀ (b : β), TopologicalSemiring (C b)] :
              TopologicalSemiring ((b : β) → C b)
              instance Pi.instTopologicalRing {β : Type u_2} {C : βType u_3} [(b : β) → TopologicalSpace (C b)] [(b : β) → NonUnitalNonAssocRing (C b)] [∀ (b : β), TopologicalRing (C b)] :
              TopologicalRing ((b : β) → C b)
              theorem TopologicalRing.of_addGroup_of_nhds_zero {R : Type u_2} [NonUnitalNonAssocRing R] [TopologicalSpace R] [TopologicalAddGroup R] (hmul : Filter.Tendsto (Function.uncurry fun x x_1 => x * x_1) (nhds 0 ×ˢ nhds 0) (nhds 0)) (hmul_left : ∀ (x₀ : R), Filter.Tendsto (fun x => x₀ * x) (nhds 0) (nhds 0)) (hmul_right : ∀ (x₀ : R), Filter.Tendsto (fun x => x * x₀) (nhds 0) (nhds 0)) :
              theorem TopologicalRing.of_nhds_zero {R : Type u_2} [NonUnitalNonAssocRing R] [TopologicalSpace R] (hadd : Filter.Tendsto (Function.uncurry fun x x_1 => x + x_1) (nhds 0 ×ˢ nhds 0) (nhds 0)) (hneg : Filter.Tendsto (fun x => -x) (nhds 0) (nhds 0)) (hmul : Filter.Tendsto (Function.uncurry fun x x_1 => x * x_1) (nhds 0 ×ˢ nhds 0) (nhds 0)) (hmul_left : ∀ (x₀ : R), Filter.Tendsto (fun x => x₀ * x) (nhds 0) (nhds 0)) (hmul_right : ∀ (x₀ : R), Filter.Tendsto (fun x => x * x₀) (nhds 0) (nhds 0)) (hleft : ∀ (x₀ : R), nhds x₀ = Filter.map (fun x => x₀ + x) (nhds 0)) :

              In a topological semiring, the left-multiplication AddMonoidHom is continuous.

              In a topological semiring, the right-multiplication AddMonoidHom is continuous.

              instance Subring.instTopologicalRing {α : Type u_1} [TopologicalSpace α] [Ring α] [TopologicalRing α] (S : Subring α) :
              TopologicalRing { x // x S }

              The (topological-space) closure of a subring of a topological ring is itself a subring.

              Instances For
                theorem Subring.topologicalClosure_minimal {α : Type u_1} [TopologicalSpace α] [Ring α] [TopologicalRing α] (s : Subring α) {t : Subring α} (h : s t) (ht : IsClosed t) :
                def Subring.commRingTopologicalClosure {α : Type u_1} [TopologicalSpace α] [Ring α] [TopologicalRing α] [T2Space α] (s : Subring α) (hs : ∀ (x y : { x // x s }), x * y = y * x) :

                If a subring of a topological ring is commutative, then so is its topological closure.

                Instances For

                  Lattice of ring topologies #

                  We define a type class RingTopology α which endows a ring α with a topology such that all ring operations are continuous.

                  Ring topologies on a fixed ring α are ordered, by reverse inclusion. They form a complete lattice, with the discrete topology and the indiscrete topology.

                  Any function f : α → β induces coinduced f : TopologicalSpace α → RingTopology β.

                  structure RingTopology (α : Type u) [Ring α] extends TopologicalSpace , TopologicalRing :

                    A ring topology on a ring α is a topology for which addition, negation and multiplication are continuous.

                    Instances For
                      theorem RingTopology.toTopologicalSpace_injective {α : Type u_1} [Ring α] :
                      Function.Injective RingTopology.toTopologicalSpace
                      theorem RingTopology.ext {α : Type u_1} [Ring α] {f : RingTopology α} {g : RingTopology α} (h : TopologicalSpace.IsOpen = TopologicalSpace.IsOpen) :
                      f = g

                      The ordering on ring topologies on the ring α. t ≤ s if every set open in s is also open in t (t is finer than s).

                      Ring topologies on α form a complete lattice, with the discrete topology and the indiscrete topology.

                      The infimum of a collection of ring topologies is the topology generated by all their open sets (which is a ring topology).

                      The supremum of two ring topologies s and t is the infimum of the family of all ring topologies contained in the intersection of s and t.

                      def RingTopology.coinduced {α : Type u_2} {β : Type u_3} [t : TopologicalSpace α] [Ring β] (f : αβ) :

                      Given f : α → β and a topology on α, the coinduced ring topology on β is the finest topology such that f is continuous and β is a topological ring.

                      Instances For
                        theorem RingTopology.coinduced_continuous {α : Type u_2} {β : Type u_3} [t : TopologicalSpace α] [Ring β] (f : αβ) :

                        The forgetful functor from ring topologies on a to additive group topologies on a.

                        Instances For

                          The order embedding from ring topologies on a to additive group topologies on a.

                          Instances For
                            def AbsoluteValue.comp {R : Type u_1} {S : Type u_2} {T : Type u_3} [Semiring T] [Semiring R] [OrderedSemiring S] (v : AbsoluteValue R S) {f : T →+* R} (hf : Function.Injective f) :

                            Construct an absolute value on a semiring T from an absolute value on a semiring R and an injective ring homomorphism f : T →+* R

                            Instances For