As stated in the famous textbook of Homological Algebra[Cartan1956], the tensor product $ A \otimes_\Lambda C$ defined between a right $ \Lambda$ -module $ A$ and a left $ \Lambda$ -module $ C$ is a quotient group $ F/R$ , where $ F$ is the free abelian group generated by $ (a,c) \in A \times C$ and $ R$ is its subgroup generated by elements of the form $ (a+a^\prime,c)-(a,c)-(a^\prime,c) $ , $ (a, c+c^\prime)-(a,c)-(a,c^\prime) $ , $ (a\lambda,c) – (a,\lambda c) $
That means, $ R$ is a representation of some kinds of equivalence, by the natural mapping $ F \to F/R = A \otimes_\Lambda C$ , this equivalence is reduced.
The question is, why we generate $ R$ using the elements of these forms? They look like some results from concrete calculations. Thanks very much if you can give some hints or references about how those generating elements come from, especially the historic problems that attracted the attention of the early mathematicians to study and developped such results.