Let $x,y,z,t$ be elements of a ring $R$ such that $xz=yt=1,xt=yz=0$ and $zx+ty=1$. Prove that the left $R$-modules $R$ and $R \oplus R$ are isomorphic

Question

this is a question that comes from UCLA algebra qualifying 2005 fall. I have been confused for two days but have no idea how to show this isomorphism. Obviously, $R$ is not a commutative ring.

Answer 1

Thanks to @Mariano Suárez-Álvarez 's suggestion, I found it amazing to block R by matrix. Assuming we have matrix $A: R\oplus R \rightarrow R$ by $\left [ \begin{matrix} c & d \end{matrix} \right ] *\left [ \begin{matrix} r_1 \\ r_2 \\ \end{matrix} \right ] $ where $(r_1,r_2) \in R$ and $B: R \rightarrow R\oplus R$ by $\left [ \begin{matrix} a \\ b \\ \end{matrix} \right ] *r $ where $r \in R.$ Then by composition of the two maps we must have $AB=Id_R$ and $BA=Id_{R\oplus R}$ Compare the coefficient in the matrix, we have A= $\left [ \begin{matrix} z & t \end{matrix} \right ]$ and $B=\left [ \begin{matrix} x \\ y \\ \end{matrix} \right ]$