= Disconnected reduced spectrum product decomposition
{title2=$R\cong R/I\times R/J$}
For a <reduced ring> with a nonempty disconnected <spectrum of a commutative ring>, choose a separation $V(I)\sqcup V(J)$. Disjointness gives $I+J=R$; covering the spectrum gives $IJ\subseteq\sqrt{(0)}=0$. For <comaximal ideals>, $I\cap J=IJ$. The <Chinese remainder theorem> then supplies a product of two nonzero quotient <rings>. Equivalently, if $i+j=1$ with $i\in I$, $j\in J$, then $i,j$ are nontrivial orthogonal <idempotents>. This connects topology of the spectrum to <ring product decomposition by an idempotent> without a Noetherian assumption.
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