Tensor Products of Free Modules (Algebra 2: Lecture 21 Video 4)

Lecture 21: We started this lecture by proving a result about spanning sets of tensor products of modules. We then saw that m \otimes 0 = 0 \otimes n = 0. We proved that for any finite abelian group A we have Q \otimes_Z A = 0, and in fact, saw something a little more general than that. We discussed how to understand R-module homomorphisms from a tensor product and emphasized that these maps come from bilinear functions from M x N. We discussed some basic questions about tensor products, such as, "What does it mean to say that m \otimes n = 0?” and, "What does it mean to say that M \otimes_R N = 0?” We discussed the tensor product of two finite cyclic groups and saw how to give a basis for the tensor product of two free R-modules. We gave an example of a tensor in the tensor product of a free R-module with itself that we could prove was not an elementary tensor. At the end of the lecture we stated some results that you should be aware of, but we did not give full proofs.