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What I understand so far:

I understand the definitions of vector space, modules, algebras. I am also acquainted with basic properties of Tensor Product of Vector spaces, especially the "universal property" which I am told is the most important thing. My knowledge of tensor products basically ends here.

For Tensor Products of Modules and Algebras, are they very different things from Tensor Product of vector spaces? Or is there a strong similarity? Is there a succinct way to summarize their similarities and differences?

Thanks.

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Tensor product of vector spaces and algebras are both special cases of tensor product of modules. Indeed a vector space is just a module over some field and an algebra is just a module that has a ring structure.

So you really need to understand the tensor product of modules to get the big picture. To make things easy it is better to focus on commutative rings, so that you don't have to care about left-right modules. Then the tensor product is defined with the same universal property has in the vector space case.

Then, as I said, the tensor product of vector space will be the special case where your ring is a field.

For algebras, the interesting thing is that the tensor product of two algebras over some ring is again an algebra over this ring. This you can prove using only the universal property of the tensor product.

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