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

free module over a set Let be any set, and let be a ring. Then the free module is defined to be the module together with the module operation and the obvious addition . 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: . 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 . 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 associativity of the tensor product Let be a ring