Lemma 10.21.4. Let $R$ be a nonzero ring. Then $\mathop{\mathrm{Spec}}(R)$ is connected if and only if $R$ has no nontrivial idempotents.
Proof. Obvious from Lemma 10.21.3 and the definition of a connected topological space. $\square$
Lemma 10.21.4. Let $R$ be a nonzero ring. Then $\mathop{\mathrm{Spec}}(R)$ is connected if and only if $R$ has no nontrivial idempotents.
Proof. Obvious from Lemma 10.21.3 and the definition of a connected topological space. $\square$
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