Lemma 10.24.3. Let $R$ be a ring. If $\mathop{\mathrm{Spec}}(R) = U \amalg V$ with both $U$ and $V$ open then $R \cong R_1 \times R_2$ with $U \cong \mathop{\mathrm{Spec}}(R_1)$ and $V \cong \mathop{\mathrm{Spec}}(R_2)$ via the maps in Lemma 10.21.2. Moreover, both $R_1$ and $R_2$ are localizations as well as quotients of the ring $R$.

Proof. By Lemma 10.21.3 we have $U = D(e)$ and $V = D(1-e)$ for some idempotent $e$. By Lemma 10.24.2 we see that $R \cong R_ e \times R_{1 - e}$ (since clearly $R_{e(1-e)} = 0$ so the glueing condition is trivial; of course it is trivial to prove the product decomposition directly in this case). The lemma follows. $\square$

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