Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 2 / 1E / ii / Solution

Conversely, if $R\cong R_1\times R_2$ with both factors nontrivial, the inverse image of $(1_{R_1},0)$ is an idempotent. It is neither zero nor one, so (i) holds. This completes the ring product decomposition by an idempotent equivalence.