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EDIT: The original question had a trivial answer: it's just a coproduct. New question below New Question: As shown below, in the category of commutative unital rings, the coproduct of a ring $R$ with $...

Call a nonzero object of a pointed category simple if it has no proper quotients, and indecomposable if it's not the product of two objects (dual to connected). Idempotents seem to pop up in many ...