Let $k$ be a field. I want to check that $$k\langle x,y \mid xy = yx = 1\rangle \cong k[u,u^{-1}],$$by defining the isomorphism explictly. Here $k[u,u^{-1}]$ denotes the ring of polynomials in the (commutative) variables $u$ and $u^{-1}$.
I start defining $j:\{x,y\} \to k[u,u^{-1}]$ putting $j(x) = u$ and $j(y) = u^{-1}$. By universality we get a ring homomorphism $\varphi :k\langle x,y\rangle \to k[u,u^{-1}]$ extending $j$. That $\varphi$ satisfies $$\varphi(xy) = \varphi(x)\varphi(y) = uu^{-1}= 1 \quad \mbox{and}\quad \varphi(yx) = \varphi(y)\varphi(x) = u^{-1}u = 1,$$so it passes to the quotient as $\overline{\varphi}\colon k\langle x,y \mid xy=yx = 1\rangle \to k[u,u^{-1}]$. I claim that $\overline{\varphi}$ is a ring isomorphism. Let's give the inverse: $\psi\colon k[u,u^{-1}] \to k\langle x,y\mid xy=yx=1\rangle$, with $$\psi\left(\sum_{r=1}^n a_r u^r + \sum_{s=1}^mb_su^{-s} \right) = \sum_{r=1}^n a_r\overline{x}^r + \sum_{s=1}^m b_s\overline{y}^s,$$where $\overline{x}$ and $\overline{y}$ are the classes of $x$ and $y$ modulo the relations given. We have $$\begin{align} \overline{\varphi} \circ \psi\left(\sum_{r=1}^n a_r u^r + \sum_{s=1}^mb_su^{-s} \right) &= \overline{\varphi}\left(\sum_{r=1}^n a_r\overline{x}^r + \sum_{s=1}^m b_s\overline{y}^s\right) \\ &= \sum_{r=1}^n a_r \overline{\varphi}(\overline{x})^r + \sum_{s=1}^m b_s \overline{\varphi}(\overline{y})^s \\ &=\sum_{r=1}^n a_r \varphi(x)^r + \sum_{s=1}^m b_s \varphi(y)^s \\ &= \sum_{r=1}^n a_r u^r + \sum_{s=1}^mb_su^{-s} ,\end{align}$$and we conclude that $\overline{\varphi}\circ \psi = {\rm Id}$.
I am having trouble checking that the other composition is $\psi \circ \overline{\varphi} = {\rm Id}$.
I don't know the explicit form of a generic element in $k\langle x,y \mid xy=yx = 1\rangle$. Is it enough to check that $$\psi \circ \overline{\varphi}(\overline{x}) = \psi(\varphi(x)) = \psi(u) = \overline{x}$$and similarly for $\overline{y}$? Yes or no, and why? Thanks for any help.
Insecure me forgot that $\overline{x}$ and $\overline{y}$ are generators for $k\langle x,y \mid xy=yx=1\rangle$ and since $\psi\circ\overline{\varphi}$ and ${\rm Id}$ are homomorphisms, the answer to my last question is "yes". Take a class in the quotient and pick a representant. Write representant as combination of $x$ and $y$. Project everything in the quotient. Done.