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.
Let $\alpha:\mathbb Q(x)\to \mathbb Q(x)$ be the field endomorphism determined by $x\mapsto x^2$, and let $A=\mathbb Q(x)[Y;\alpha]$ be the twisted polynomial ring (such that $Yq:=\alpha(q)Y$ for $q\in \mathbb Q(x)$.
For every odd integer $n$, let $\nu_n$ be the valuation on $\mathbb Q(x)$ induced by the $n$th cyclotomic polynomial. Each one can be extended to a valuation on $A$ by defining $\nu_n(\sum d_iY^i)=\min_i \nu_n(d_i)$.
Let $V$ be an infinite set of valuations of this type with the property that at any $q\in \mathbb Q(x)\setminus\{0\}$, only finitely many elements of $V$ are nonzero, and let $B$ be the intersection of valuation subrings of $A$ for valuations in $V$. $B$ is a ring which is right primitive but not left primitive.