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I'm studying tensor products right now and I've came across multiple definitions. The one I'm confused with is when we have vector spaces $V$ and $W$ and we define the tensor product as the quotient of the free module $F$ of $V \times W$ and $E$, where $E$ can either be:

Definition 1: $E$ is the subspace of $F$ spanned by all elements: $$(v_1+v_2,w)-(v_1,w)-(v_2,w)$$ $$(v,w_1+w_2)-(v,w_1)-(v,w_2)$$ $$a(v,w)-(av,w)$$ $$a(v,w)-(v,aw)$$

or definition 2: $E$ is the subspace of $F$ spanned by all elements: $$(v_1+v_2,w)-(v_1,w)-(v_2,w)$$ $$(v,w_1+w_2)-(v,w_1)-(v,w_2)$$ $$(av,w)-(v,aw)$$

Are these definitions equivalent? (namely, what I'm asking is, does: $(al_1 \otimes l_2) = (l_1 \otimes al_2), = a(l_1 \otimes l_2)$ hold in definition 2)?

Thanks very much

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1 Answer 1

None of these is a definition. The correct definition of the tensor product is the universal property: the tensor product classifies bilinear maps. Your "definitions" are actually constructions (but there are also other constructions of the tensor product; of course isomorphic). Now to answer your question: The first construction is the correct one, the second is not correct. If $\beta : V \times W \to U$ is a map which is additive in each variable and balanced in the sense that $\beta(av,w)=\beta(v,aw)$ for $a \in K$, $v \in V$ and $w \in W$, then there is no reason why we can conclude $a \beta(v,w) = \beta(a v,w)$. Try to find an example for $V=W=U=K=\mathbb{C}$.

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Actually the second construction is the correct one in the context of noncommutative rings and bimodules. If someone is interested, the Wikipedia article on tensor products explains this. –  Martin Brandenburg Dec 1 '13 at 9:57
    
I just read the wiki article. Would you care elaborating? The problem I'm having is that, this the 2nd definition, nothing ever "goes outside" the tensor product. So is the basis the tensors of all the basis of V and W? What if you have the 2nd definition but the underlying ring is commutative anyway? etc –  user112532 Dec 1 '13 at 11:15
    
I don't understand the question. You can find everything in the Wikipedia article, and my answer also gives a hint how to find the difference between the two constructions. –  Martin Brandenburg Dec 1 '13 at 19:24
    
Thanks for your help with this. The issue I am having is that some notions of the two constructions seem too different. For example the proof (at least the one I've seen) that if $v_1,v_2...$ is a basis for V and $w_1,w_2,...$ is a basis for $W$ then the tensors $v_iw_j$ are a basis for the tensor product seem to not work (namely we can't express any tensor in terms of linear combinations of the $(v_i \otimes w_j)$ because nothing "goes outside" the tensors as in definition 1)? If not, can you direct me to a proof? –  user112532 Dec 1 '13 at 23:34
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