Let $V$ be a finite dimensional vector space over a field $K$ of characteristic zero. Assume that we are given a set of (not necessarily homogeneous) elements $f_1,\ldots f_n$ in the tensor algebra $T(V)$. How can we decide if the ideal $I$ generated by the elements $f_i$ is the entire ring $T(V)$ or not? Is there any particular algorithm which is suitable for this problem? I am not interested in the particular structure of the ring $T(V)/I$, but only in the question of whether or not it is trivial. Another related question: how can we decide if $T(V)/I$ is finite dimensional or not?
It is well known that the analogous question is undecideable for Groups. (By Encoding a Turing machine into the Generators and relations and using undecideablity theorems like the undecideabilty of the halting Problem there.)
You can expect that the question for algebras is undecideable as well, as you can for example encode Groups (or Turing machines) in your Algebra.