In algebra, the tensor product is a way to construct a new module from two given modules, effectively allowing us to "multiply" the modules together. It is particularly useful in the context of linear algebra, representation theory, and algebraic topology. ### Definition Let \( R \) be a ring, and let \( M \) and \( N \) be two \( R \)-modules.
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The tensor product of an -module and an -module is an -module generated by pure tensors , subject to additivity in both variables and the balancing relation