Every isomorphism of commutative ring with 1 with product of two rings is defined by idempotent element $x$ and element $1-x$.

Question

I proved that $f: R \rightarrow P_x \times P_{x\prime}$ where $x$ is impodent element in $R$, $x\prime=(1-x)$ (also impodent) and $P_x=\{rx:r \in R\}$. But I have problem with proving that every isomorphism of ring with product of two rings is defined by idempotent element and it's opposite element.