$\DeclareMathOperator\la{\langle}\DeclareMathOperator\ra{\rangle}$Let $K$ be a field an $K\la x,y,z\ra$ the non-commutative polynomial ring in 3 variables.

Question 1: Are there three (less is probably not possible?!) polynomials $f,g,h \in K\la x,y,z\ra$, which are sums of monomials of degree at least two such that the algebra $K\la x,y,z\ra/(f,g,h)$ is finite dimensional?

Is there a systematic way to construct such polynomials?

For two variables we can take for example $f=xy+yx$ and $g=x^3+y^3$, but I do not know (or forgot) such examples for more than 2 variables.

(If possible we should also have (this is equivalent to $(f,g,h)$ being an admissible ideal) that there exists an $n \geq 2$ such that $J^n \subseteq (f,g,h)$ where $J=\la x,y,z\ra$ is the ideal generated by x,y,z.)

Question 2: Let $f_i$ for $i=1,...,n-1$ be polynomials spanning an admissible ideal in the non-commutative polynomial ring in $n-1$ variables $x_i$. Can we find $g$ such that $\la f_i,g\ra$ is an admissible ideal in $K\la x_i,y\ra$?

$\newcommand\la{\langle}\newcommand\ra{\rangle}$Let $K$ be a field and $K\la x,y,z\ra$ the non-commutative polynomial ring in 3 variables.

Question 1: Are there three (fewer is probably not possible?!) polynomials $f,g,h \in K\la x,y,z\ra$, which are sums of monomials of degree at least two, such that the algebra $K\la x,y,z\ra/(f,g,h)$ is finite dimensional?

Is there a systematic way to construct such polynomials?

For two variables we can take for example $f=xy+yx$ and $g=x^3+y^3$, but I do not know (or forgot) such examples for more than 2 variables.

(If possible we should also have (this is equivalent to $(f,g,h)$ being an admissible ideal) that there exists an $n \geq 2$ such that $J^n \subseteq (f,g,h)$ where $J=\la x,y,z\ra$ is the ideal generated by $x,y,z$.)

Question 2: Let $f_i$ for $i=1,\dotsc,n-1$ be polynomials spanning an admissible ideal in the non-commutative polynomial ring in $n-1$ variables $x_i$. Can we find $g$ such that $\la f_i,g\ra$ is an admissible ideal in $K\la x_i,y\ra$?

Let $K$ be a field an $K<x,y,z>$ the non-commutative polynomial ring in 3 variables.

Question 1: Are there three (less is probably not possible?!) polynomials $f,g,h \in K<x,y,z>$, which are sums of monomials of degree at least two such that the algebra $K<x,y,z>/(f,g,h)$ is finite dimensional?

Is there a systematic way to construct such polynomials?

For two variables we can take for example $f=xy+yx$ and $g=x^3+y^3$, but I do not know (or forgot) such examples for more than 2 variables.

(If possible we should also have (this is equivalent to $(f,g,h)$ being an admissible ideal) that there exists an $n \geq 2$ such that $J^n \subseteq (f,g,h)$ where $J=<x,y,z>$ is the ideal generated by x,y,z.)

Question 2: Let $f_i$ for $i=1,...,n-1$ be polynomials spanning an admissible ideal in the non-commutative polynomial ring in $n-1$ variables $x_i$. Can we find $g$ such that $<f_i,g>$ is an admissible ideal in $K<x_i,y>$?

$\DeclareMathOperator\la{\langle}\DeclareMathOperator\ra{\rangle}$Let $K$ be a field an $K\la x,y,z\ra$ the non-commutative polynomial ring in 3 variables.

Question 1: Are there three (less is probably not possible?!) polynomials $f,g,h \in K\la x,y,z\ra$, which are sums of monomials of degree at least two such that the algebra $K\la x,y,z\ra/(f,g,h)$ is finite dimensional?

Is there a systematic way to construct such polynomials?

For two variables we can take for example $f=xy+yx$ and $g=x^3+y^3$, but I do not know (or forgot) such examples for more than 2 variables.

(If possible we should also have (this is equivalent to $(f,g,h)$ being an admissible ideal) that there exists an $n \geq 2$ such that $J^n \subseteq (f,g,h)$ where $J=\la x,y,z\ra$ is the ideal generated by x,y,z.)

Question 2: Let $f_i$ for $i=1,...,n-1$ be polynomials spanning an admissible ideal in the non-commutative polynomial ring in $n-1$ variables $x_i$. Can we find $g$ such that $\la f_i,g\ra$ is an admissible ideal in $K\la x_i,y\ra$?

Let $K$ be a field an $K<x,y,z>$ the non-commutative polynomial ring in 3 variables.

Question 1: Are there two polynomials $f,g \in K<x,y,z>$, which are sums of monomials of degree at least two such that the algebra $K<x,y,z>/(f,g)$ is finite dimensional?

For two variables we can take for example $f=xy+yx$ and $g=x^3+y^3$, but I do not know (or forgot) such examples for more than 2 variables.

(If possible we should also have (this is equivalent to $(f,g)$ being an admissible ideal) that there exists an $n \geq 2$ such that $J^n \subseteq (f,g)$ where $J=<x,y,z>$ is the ideal generated by x,y,z.)

Question 2: Let $f_i$ for $i=1,...,n-1$ be polynomials spanning an admissible ideal in the non-commutative polynomial ring in $n-1$ variables $x_i$. Can we find $g$ such that $<f_i,g>$ is an admissible ideal in $K<x_i,y>$?

Let $K$ be a field an $K<x,y,z>$ the non-commutative polynomial ring in 3 variables.

Question 1: Are there three (less is probably not possible?!) polynomials $f,g,h \in K<x,y,z>$, which are sums of monomials of degree at least two such that the algebra $K<x,y,z>/(f,g,h)$ is finite dimensional?

Is there a systematic way to construct such polynomials?

For two variables we can take for example $f=xy+yx$ and $g=x^3+y^3$, but I do not know (or forgot) such examples for more than 2 variables.

(If possible we should also have (this is equivalent to $(f,g,h)$ being an admissible ideal) that there exists an $n \geq 2$ such that $J^n \subseteq (f,g,h)$ where $J=<x,y,z>$ is the ideal generated by x,y,z.)

Question 2: Let $f_i$ for $i=1,...,n-1$ be polynomials spanning an admissible ideal in the non-commutative polynomial ring in $n-1$ variables $x_i$. Can we find $g$ such that $<f_i,g>$ is an admissible ideal in $K<x_i,y>$?