Linear Algebra over a Ring/Multilinear algebra

multilinear function Let be a ring, and let be -modules. Then the set of -multilinear functions from to is the set . equivalent definition of tensor product of free modules using multilinear functions Let be a ring, and let be free, finitely generated -modules. Then if we alternatively define , and let the elementary tensors be , then the from this definition satisfies the same universal property as the usual tensor product . In particular, the two tensor products are canonically isomorphic. For , let be a basis of , where is the respective finite index set. Given any -module and any multilinear map , we want a unique linear function such that , where is the map that sends a tuple to the respective elementary tensor.