Examples
Consider the rational numbers Q and the integers modulo n Zn. As with any abelian group, both can be considered as modules over the integers, Z. Let B: Q × Zn → M be a Z-bilinear operator. Then B(q, k) = B(q/n, nk) = B(q/n, 0) = 0, so every bilinear operator is identically zero. Therefore, if we define to be the trivial module, and to be the zero bilinear function, then we see that the properties for the tensor product are satisfied. Therefore, the tensor product of Q and Zn is {0}.
An abelian group is a Z-module, which allows the theory of abelian groups to be subsumed in that of modules. The tensor product of Z-modules is sometimes termed the tensor product of abelian groups.
Read more about this topic: Tensor Product Of Modules
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