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Mathlib.Topology.Algebra.Ring.Ideal

Ideals and quotients of topological rings #

In this file we define Ideal.closure to be the topological closure of an ideal in a topological ring. We also define a TopologicalSpace structure on the quotient of a topological ring by an ideal and prove that the quotient is a topological ring.

def Ideal.closure {R : Type u_1} [TopologicalSpace R] [Ring R] [TopologicalRing R] (I : Ideal R) :

The closure of an ideal in a topological ring as an ideal.

Equations
  • I.closure = { carrier := closure I, add_mem' := , zero_mem' := , smul_mem' := }
Instances For
    @[simp]
    theorem Ideal.coe_closure {R : Type u_1} [TopologicalSpace R] [Ring R] [TopologicalRing R] (I : Ideal R) :
    I.closure = closure I
    theorem Ideal.closure_eq_of_isClosed {R : Type u_1} [TopologicalSpace R] [Ring R] [TopologicalRing R] (I : Ideal R) (hI : IsClosed I) :
    I.closure = I
    Equations
    Equations
    • =