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
Tensor product need not commute with an infinite direct product. For example,The element has infinite order in the product, so it survives rational localization.
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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.