This is the description directly from A Ring Primitive On The Right But Not On The Left by G. M. Bergman, published in Proceedings of the American Mathematical Society 15.3 (1964): 473-475.
See O'Meara's construction of $R_{81}$ and $R_{82}$ at the end of the description of $R_{80}$.
Nielsen and S̆ter construct it with infinite matrices in Section 2 of Connections between unit-regularity, regularity, cleanness, and strong cleanness of elements and rings, but they also note their construction is isomorphic to the opposite ring of Bergman's original construction.
Following Goodearl: let $F$ be a field, $T=F[[t]]$ and $K=F((t))$. Let $V$ be the subspace of $K$ generated by the negative powers of $t$. Then $K=V\oplus T$. Let $p$ denote the projection in $End(K_F)$ projecting onto $V$ in this decomposition.
Set $R_1=\{x\in End(K_F)\mid x(V)\subseteq V, x(T)\subseteq T\}$. Set $R_2=\{x\in End(K_F)\mid \exists a\in K,n\in\mathbb N^+ \text{ such that } (x-a)(t^nT)=0\text{ and } (x-a)K\subseteq t^{-n}T\}$. $R=R_1\cap R_2$.
For each $x\in R$, we can define $\psi(x)=a$ where $(x-a)(t^nT)=0\text{ and } (x-a)K\subseteq t^{-n}T$ from $R\to End(K_F)$. Define $J=\{x\in End(K_F)\mid \psi(x)=0 \text{ and } dim(xK)<\infty\}$. It turns out that the image of $\psi$ restricted to $R$ is exactly $R_{70}$.
$Q = R + J$ is a subring of $R_2$, and $Q$ is $R_{81}$ and $R$ is $R_{82}$.