Algebraic Structure with Duality between Addition and Multiplication
As I understand it, you're looking for a set $A$ with two operations $\cdot$ and $+$ such that $(R, +, \cdot)$ is a ring and $(R, \cdot, +)$ is a ring.
Unfortunately, this can't really happen. It's not hard to show that in any ring, $0 \cdot r = 0$ for all $r \in R$. So $1 + r = 1$ for all $r \in R$ also, since $(R, \cdot, +)$ is a ring. But this implies $r = 0$ for all $r \in R$, so the only example is the trivial ring.