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| /- | |
| Copyright (c) 2021 María Inés de Frutos-Fernández. All rights reserved. | |
| Released under Apache 2.0 license as described in the file LICENSE. | |
| Authors: María Inés de Frutos-Fernández | |
| -/ | |
| import ring_theory.localization.basic | |
| import topology.algebra.ring | |
| /-! | |
| # Localization of topological rings | |
| The topological localization of a topological commutative ring `R` at a submonoid `M` is the ring | |
| `localization M` endowed with the final ring topology of the natural homomorphism sending `x : R` | |
| to the equivalence class of `(x, 1)` in the localization of `R` at a `M`. | |
| ## Main Results | |
| - `localization.topological_ring`: The localization of a topological commutative ring at a submonoid | |
| is a topological ring. | |
| -/ | |
| variables {R : Type*} [comm_ring R] [topological_space R] {M : submonoid R} | |
| /-- The ring topology on `localization M` coinduced from the natural homomorphism sending `x : R` | |
| to the equivalence class of `(x, 1)`. -/ | |
| def localization.ring_topology : ring_topology (localization M) := | |
| ring_topology.coinduced (localization.monoid_of M).to_fun | |
| instance : topological_space (localization M) := | |
| localization.ring_topology.to_topological_space | |
| instance : topological_ring (localization M) := | |
| localization.ring_topology.to_topological_ring | |