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I think this problem is a simple generalization of a standard factorization theorem over the integers formulated in the language of abstract algebra but I have not quite been able to generalize the proof.

Let $V$ be a finite dimensional vector space

Let $T(V) = \oplus_{k=0}^{\infty} T^{k}(V)$ where $T^k(V) = V \otimes V \otimes \ldots \otimes V$ is tensor product of $k$ vector spaces.

$T(V)$ is an algebra and so has a multiplication operation defined by the canonical isomorphism $T^kV \otimes T^\ell V \to T^{k + \ell}V $ (see Wikipedia).

Suppose there exists two elements $f,g \in T(V)$ as well as elements $x,y \in T(V)$ such that $f \otimes x =g \otimes y \neq 0$

How do we show one of the elements $f,g$ is a right multiple of the other?

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Do you mean $f\otimes x=g\otimes y$? If not, what is this multiplication? Why do you introduce $V$ as a vector space and then speak of the tensor product of modules? The "both" doesn't really make sense, since these two are equal; it would be clearer to just write "such that $fx=gy\ne0$". But most importantly, what standard factorization theorem over the integers are you referring to? $14\cdot15=10\cdot21$, but $14$ and $10$ aren't multiples of each other. –  joriki Oct 20 '11 at 6:24
    
@joriki - I was under the impression the multiplication operation in the algebra was defined by the isomorphsim TkV⊗TℓV→Tk+ℓV and that this was different than the tensor product, is this wrong? –  user7980 Oct 20 '11 at 6:49
    
You're right, technically, but the notation with $\otimes$ is often used for the multiplication that you mean. I was just checking whether this is what you mean since you hadn't introduced any multiplication, or introduced $T(V)$ as an algebra, at that point. But what about my other comments? I think the question would make a lot more sense if you explicate which "standard factorization theorem over the integers" you're referring to. –  joriki Oct 20 '11 at 7:53
    
@joriki - Thanks for your help. As for your last comment I was just thinking of ideas for a proof but your example clearly shows that we cant use properties of the integers to get a feel for the proof of the general result. –  user7980 Oct 20 '11 at 21:28
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