Cardinality of localizations

In this file, we establish the cardinality of localizations. In most cases, a localization has cardinality equal to the base ring. If there are zero-divisors, however, this is no longer true - for example, ZMod 6 localized at {2, 4} is equal to ZMod 3, and if you have zero in your submonoid, then your localization is trivial (see IsLocalization.uniqueOfZeroMem).

Main statements

• IsLocalization.card_le: A localization has cardinality no larger than the base ring.
• IsLocalization.card: If you don't localize at zero-divisors, the localization of a ring has cardinality equal to its base ring,

theorem IsLocalization.card_le {R : Type u} [CommRing R] (S : Submonoid R) {L : Type u} [CommRing L] [Algebra R L] [IsLocalization S L] : Cardinal.mk L ≤ Cardinal.mk R

A localization always has cardinality less than or equal to the base ring.

theorem IsLocalization.card {R : Type u} [CommRing R] (S : Submonoid R) (L : Type u) [CommRing L] [Algebra R L] [IsLocalization S L] (hS : S ≤ nonZeroDivisors R) : Cardinal.mk R = Cardinal.mk L

If you do not localize at any zero-divisors, localization preserves cardinality.