Let $K \langle x , y \rangle $ ($K$ a field, we can assume it has only two elements if it helps) be the non-commutative polynomial ring in 2 variables.
Question 1: For which non-commutative polynomials $f$ in two variables is the $K$-algebra $A_f=K \langle x , y \rangle / \langle x^2 ,f \rangle$ finite dimensional and the ideal $I_f=\langle x^2 ,f \rangle$ admissible?
Here admissible means that $J^n \subseteq I_f \subseteq J^2$ for some $n \geq 2$ and where $J=\langle x,y \rangle $ is the ideal spanned by $x,y$ in $K \langle x , y \rangle $.
Thus we can assume that $f$ does not contains $x^2$ and is a sum of non-commutative monomials of degree at least 2.
For example $f=xy+y^2$ works and gives a 6-dimensional algebra, while $f=xy+yx+y^2$ gives an infinite dimensional algebra and $f=yx+xy^2+y^3$ gives a finite dimensional but non-admissible example. I tried some experiments with GAP but checking whether an algebra is finite dimensional seems to take forever for big polynomials $f$.
