universal property of a ring localization

1. Proposition

Let 20230724-universal_property_of_a_localization_6130b692b4d58a44d70e323b9432701598ebebee.svg be a commutative ring, 20230724-universal_property_of_a_localization_c2f0e20f9fbd324ac350ac70921d96e3b8b9b06f.svg a localization along 20230724-universal_property_of_a_localization_1ff99e456c0797b98eaacbda9c1dffef2e88d53e.svg, 20230724-universal_property_of_a_localization_fc320dcdb38675d7480f71f77d5579cca1b56b4c.svg the canonical ring homomorphism for a localization and

20230724-universal_property_of_a_localization_9f55fae4c8c5e72fa58a93a3977ef33245c5ff74.svg

a ringhomomorphism, such that 20230724-universal_property_of_a_localization_dae63817787b26e69a12b96c70095ba140b7625d.svg Then there exists a unique homomorphism 20230724-universal_property_of_a_localization_1a35d57ae086edbcfbe850b50dd91978a943c540.svg such that

20230724-universal_property_of_a_localization_f9fd7635e1b244ae5f2b31ff4e87a294f3ba9f35.svg

2. Proof

2.1. uniqueness

Let 20230724-universal_property_of_a_localization_86831a2513fd9c4c43640a952f7278b1352aa985.svg, then for 20230724-universal_property_of_a_localization_aca211bb4d46e9cea499e1e2c77c471cb62e4dab.svg we get by cancelling in a localization

20230724-universal_property_of_a_localization_7d5818bd7f977c4b666cd1b348ea736b7a90a702.svg

hence 20230724-universal_property_of_a_localization_cc0a59c06a313e0cf151d0d16898c996c49ccf6b.svg

Thus the map is - if existent - uniquely defined by

20230724-universal_property_of_a_localization_e80682e999d3fa7b843830b342963a9463c13a66.svg

2.2. existence

2.2.1. welldefined map

Let 20230724-universal_property_of_a_localization_b7c88ee447342cb065772535077276f6370a1ba3.svg, then by construction there exists an 20230724-universal_property_of_a_localization_86831a2513fd9c4c43640a952f7278b1352aa985.svg such that

20230724-universal_property_of_a_localization_57ae088011ba03692e68e0f07d9e00921ea058d6.svg

Therefore applying 20230724-universal_property_of_a_localization_3f6e0f644920acc65654856391d4b66324d57028.svg

20230724-universal_property_of_a_localization_81acc5da13e3bbb7cf47454806e30558602b2005.svg

By assumption 20230724-universal_property_of_a_localization_60d69de7627f1d0c2fe5f66a0e5f49f5d88fc2f1.svg hence since units are no zero divisor we conclude, that

20230724-universal_property_of_a_localization_dadbedc31368cc870e77faf558c88756772b3277.svg

Furthermore, by assumption there exist inverse elements 20230724-universal_property_of_a_localization_8b3c6315d021effaa33efd9d3455ab0cb9045570.svg, hence

20230724-universal_property_of_a_localization_e07cedcf4ee047bab6685f4151d55133003a6ed3.svg

2.2.2. ringhomomorphism

Let 20230724-universal_property_of_a_localization_6cd154312b948716a855f781065351d55973bca4.svg, then

20230724-universal_property_of_a_localization_80951387b8c521ba8151438bb03c61665c698fba.svg

or

20230724-universal_property_of_a_localization_b0e89000010ae6f3b18f3bec8e4370db9632ff80.svg

Date: nil

Author: Anton Zakrewski

Created: 2024-10-13 So 15:23