Are there any examples of affine schemes which is connected but not irreducible and not reduced?. Reducedness and irredubility doesn't deduces connectedness, so I think there should be examples, but I cannot give specific examples... Thank you for your help!
2026-03-26 03:12:48.1774494768
Affine scheme which is connected but not irreducible and not reduced.
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You want a ring which has no nontrivial idempotents, has more than one minimal prime ideal and has some nilpotents.
First, connected but not irreducible: we can take $\mathbb Z[X]/(2X)$. It has no nontrivial idempotents, and it has the minimal prime ideals $(2)$ and $(X)$.
To make it not-reduced, we can change it to for example $\mathbb Z[X]/(4X)$, or $\mathbb Z[X,Y]/(2X,Y^2)$.