Quotient of noncommutative algebra

Question

Let $R=k\langle x,y,z \rangle$ be the non-commutative algebra in $3$ variables. Let $I$ be the ideal defined by the relations $xy-yx,yz-zy,xz-zx$. How to show formally that $R/I$ is the polynomial ring $k[x,y,z]$?

Answer 1

Consider the following $k$ algebra morphism $$\psi:k\langle X,Y,Z\rangle\longrightarrow k[x,y,z] $$ $$X\mapsto x$$ $$Y\mapsto y$$ $$Z\mapsto z$$

then show that $ker \psi= I$, and by the first isomorphism theorem (of rings) we are done.