Cardinality of localizations #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

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 is_localization.unique_of_zero_mem).

Main statements #

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

source

theorem is_localization.card_le {R : Type u} [comm_ring R] (S : submonoid R) {L : Type u} [comm_ring L] [algebra R L] [is_localization S L] :

cardinal.mk L ≤ cardinal.mk R

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

source

theorem is_localization.card {R : Type u} [comm_ring R] (S : submonoid R) (L : Type u) [comm_ring L] [algebra R L] [is_localization S L] (hS : S ≤ non_zero_divisors R) :

cardinal.mk R = cardinal.mk L

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