uniqueness of a tensor product

1. Proposition

Let 20230801-uniqueness_of_a_tensor_product_6130b692b4d58a44d70e323b9432701598ebebee.svg be a ring and 20230801-uniqueness_of_a_tensor_product_2273b15653e62f647e7bff48b42dac4fad211430.svg 20230801-uniqueness_of_a_tensor_product_d9fed023a9d07263c522f5c4dbb780117ad7ad02.svg-module Then the tensor product is unique up to unique isomorphism

2. Proof

Suppose 20230801-uniqueness_of_a_tensor_product_b7ef9702ba9b88bd2e422b89d20962939f6fe211.svg are tensor products of 20230801-uniqueness_of_a_tensor_product_612a3d41795adef416750fa250ce585eadd8340d.svg and 20230801-uniqueness_of_a_tensor_product_082f9bab71de6c3b55a762c7c497e13141dc3320.svg. Then

20230801-uniqueness_of_a_tensor_product_e172fa4f6993cf270445cf4a29477b661f7a1860.svg

Hence 20230801-uniqueness_of_a_tensor_product_34f38f7642cc4de68a8d7c7e867cb3ae8b2d8f1b.svg and because of reasons of symmetry, 20230801-uniqueness_of_a_tensor_product_341117393e6aec6db80251e1baf05ef02ea853e2.svg Therefore, 20230801-uniqueness_of_a_tensor_product_85e0758714b8968697a5ccf62de8c43061fc9cf9.svg are isomorphisms

Date: nil

Author: Anton Zakrewski

Created: 2024-10-13 So 15:34