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

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- Solution ii

Solution

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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