commutativity of the tensor product for a commutative ring

1. Proposition

Let 20230803-commutativity_of_a_tensor_product_6130b692b4d58a44d70e323b9432701598ebebee.svg be a commutative ring and -modules. then

20230803-commutativity_of_a_tensor_product_9d24f8fad95f3340ff432ce046712bf849b95aa6.svg

2. Proof

Let 20230803-commutativity_of_a_tensor_product_87682205f23672547a35acda699f655af8a066e5.svg the canonical map. Let be the map into the tensor product and analogue. Then is a bilinear map Hence

20230803-commutativity_of_a_tensor_product_9313dd4a9183980baf8af0491956559ce3e12b12.svg

By reasons of symmetry we deduce, that there exist unique maps

20230803-commutativity_of_a_tensor_product_db75e00296317013d6acdcd02684f6d35d1c284c.svg

Hence

20230803-commutativity_of_a_tensor_product_431b4773e62c1fab7be2c2202e909cf71619e10f.svg