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)?