Question: Are the two sided ideals of $k\left<x,y\right>$ (polynomial ring in twonon commuting variables) finitely generated (as two sided ideals) when $k$ is a field?
I know that there are one sided ideals which are not finitely generated. Even two sided ideals are not finitely generated as one sided ones. For example simply take the two sided ideal $(x)$ it is not finitely generated as a ONE sided ideal.
I think the above is not true, but I am unable to find a counterexample.
My interest in this comes from seeing finite presentation of finitely generated algebras (non commutative) over fields. I have proved that if the algebra is finite dimensional (over $k$) then it is finitely presented. So, now I am exploring the more general case of finite generation.
I have tried taking some familiar infinite dimensional algebras and the corresponding kernels from the free algebra but there I get finitely generated kernels (as algebras even, example for any path algebra even when it has cycles and hence is infinite dimensional).
Thank you.