# Trivial algebras given by generators and relations

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?

• Thanks for the answer. Do you know if there are some results if the dimension of $V$ and the degree of the $f_i$ polynomials are restricted? In my case $V$ is of dimension 3, and there are 5 polynomials, of degree 2 (that is- containing monomials of degree at most 2) – Ehud Meir Mar 20 '16 at 20:38