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This is about problem $5$ in Section IV.$5$ of Hungerford's Algebra book. The question is the following:

If $A'$ is a submodule of the right $R$-module $A$ and $B'$ is a submodule of the left $R$-module $B$, then $$\frac{A}{A'} \otimes _R \frac{B}{B'} \cong \frac{A \otimes _R B}{C}$$ where $C$ is the subgroup of $A \otimes _R B$ generated by all elements $a' \otimes b$ and $a \otimes b'$ with $a \in A$, $a' \in A'$, $b \in B$, $b' \in B'$.

I'm just starting to somehow grasp the tensor product concept and I'm having a lot of trouble when trying to prove that some tensor product is isomorphic to a certain group.

In particular what I feel that's causing me problems is the fact that in the tensor product $A \otimes _R B$ not every element is an elementary tensor of the form $a \otimes b$ so that it is really hard for me sometimes to be able to define an appropriate map to try to produce an explicit isomorphism.

I'm not sure if the universal property could be of help here, since I can try to define a map $$\varphi : \frac{A}{A'} \times \frac{B}{B'} \longrightarrow \frac{A \otimes _R B}{C} $$ by $$\varphi(a + A', b + B') := a \otimes b + C$$

But I've been trying without getting anywhere. Any hints or advice on how to approach this exercise would be very much appreciated. Thank you.

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    Tensor product tries to linearize bilinear map. In this case, look at an arbitrary bilinear map from $A/A' \times B/B'$ to an $R$-module. If you can show that every such map can factor through $\phi$ that you constructed, then we are done. Can you lift this bilinear map to $A \times B$ so as to relate back to $A \otimes_R B$?2011-02-25
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    @Soarer I'm sorry but I think I don't fully understand the lifting part of your comment. But since the tensor product is uniquely defined up to isomorphism by the universal property, then I believe that what you mean is that I need to consider a map from $A/A' \times B/B' \longrightarrow M$ where $M$ is an arbitrary abelian group, and if I can factor this map through $\varphi$ as constructed in my question, then that would prove the isomorphism I want by uniqueness of the tensor product?2011-02-25
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    @Adrian, first some edits: 1. $R$-module should be abelian group. 2. ..every such map can factor through $\phi$ UNIQUELY.2011-02-25
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    @Adrian, yes, that's what I mean. But how can you factor through something that involves $A \otimes B$? The cleanest way is to lift your bilinear map $A/A' \times B/B' \to M$ to a bilinear map $A \times B \to M$, via the quotient maps $A \to A/A'$, $B \to B/B'$. Then by definition you can factor through the bilinear map on $A \times B$ through $A \otimes B$. The only thing you need to do now is to check this map on $A \otimes B$ factors through $A \otimes B/C$.2011-02-25
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    You say «I'm not sure if the universal property could be of help here». But you have to keep in mind that the universal property is the *only* way to define a map on a tensor product!2011-02-25
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    @Soarer Thanks, that's of great help. I'll try with this and will edit when I get all the details down or if I have problems with something. Although I think that this is enough. It would be a good idea though if you could post your comments as an answer so that people can upvote your answer an I'll have an answer to accept then.2011-02-25
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    Adrian: I wrote some course handouts on tensor products which you might find useful: http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/tensorprod.pdf and http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/tensorprod2.pdf2011-03-06
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    @KCd Thank you very much. I was just looking through your notes (the first handout at least) and they're amazing. They seem to address most of the things that I find confusing and contain lots of really nice remarks, examples and motivation. For instance the analogy between elementary tensors and PDE's is really nice. Thank you, really. I just wonder why some professors keep using old and dry books when there exist these amazing references for free online. And I'm not saying that Hungerford is a bad book, I like it, but I just don't like the module theory chapter of that book.2011-03-07
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    Adrian: Rather than wonder why your instructor (I figured out who he is by web search) sticks to sources which cover some topics in crappy ways, show him things you find useful to learn from and that may encourage him to look online for resources when he teaches the class again.2011-03-08
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    @KCd You're right. That's a really good advice. Thanks once again.2011-03-08
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    By the way, the problem you ask about in your question is Theorem 2.17 in the second handout I wrote.2011-03-09

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Given a bilinear map $A/A' \times B/B' \to M$ to some abelian group $M$, lift it to a bilinear map $A \times B \to M$. It is easy to check that

  1. $A \times B \to M$ factors through $A \otimes B/C$ uniquely,
  2. $A/A' \times B/B' \to M$ can be factored as $\phi$ and the map $A \otimes B/C \to M$ in 1 uniquely.

This verifies the universal property of tensor product.