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A question was brought up to me about if it is possible to come up with a module that has no non trivial invertible elements in its respective tensor algebra. I am not sure if this is trivial based on the following fact but I thought it would be a good starting point:

Let $T(V) = \oplus_{k=0}^{\infty} T^k(V)$ be the tensor algebra of a finite vector space $V$.

How do you show the only invertible elements in $T(V)$ are nonzero scalars (0-tensors)?

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What does "no" invertible elements mean in the first sentence? You at least have 1. –  KCd Oct 21 '11 at 6:28
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I think the direct sum should start at $k=0$? –  joriki Oct 21 '11 at 6:53
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up vote 7 down vote accepted

You can write every element as a sum of elements of individual spaces $T^k(V)$. If you multiply two non-scalar products written like that, the product of the elements of highest degrees can't cancel with anything else, so it would have to be zero for the entire product to be unity. That's not possible; hence there are no non-scalar invertible elements.

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One can rephrase this as follows: there as a very natural way to define the degree of a non-zero element of $TV$, and when one does this, one has both $\deg1=0$ and $\deg ab=\deg a+\deg b$ for all non-zero $a$, $b\in TV$. Then one can use exactly the same proof as for ordinary polynomials. –  Mariano Suárez-Alvarez Oct 23 '11 at 22:37
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