Linear Algebra over a Ring/Direct product, direct sum and tensor product

Definition (free module over a set):

Let be any set, and let be a ring. Then the free module is defined to be the module

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together with the module operation

and the obvious addition


Definition (tensor product):

Let be a ring and let be -modules. The tensor product of the modules is defined as the -module


where is the following submodule:


Proposition (universal property of the tensor product):

Let be a ring and let be -modules. Then the tensor product satisfies the universal property that for each -module and each multilinear map , there exists a unique linear map such that


{{proposition|tensor product as multifunctor|Let be a ring. Then for each , the tensor product yields a multifunctor


Whenever and are -modules and for , are morphisms, the morphisms that turn into a multifunctor are given by

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