Let $R=k\langle x,y\rangle$ be the free algebra on two variables. We can think of it as an algebra of non-commutative polynomials. Consider elements of the form $p[x,y]q$, where $p,q\in R$ are some monic monomials, and $[x,y]=xy-yx$.
My question is: how to check that elements $p[x,y]q$ are linearly independent?
I have tried to define some linear map from the subspace of $R$, spanned by all $p[x,y]q$ to some other linear space that will separate the elements, but it didn't really work. I was trying to order monomials and do some induction on the order, but it also didn't work.
I don't know the answer, so it may not be true. I might be missing something obvious, and I would be happy if you point that out for me.
Thank you very much!