 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ib/paper-2/1e/ii/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Conversely, if

R ≅ 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.

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