= Chinese remainder theorem for ideals
{c}
{title2=$R/(I_1\cdots I_r)\cong\prod_iR/I_i$}
For pairwise comaximal <ideals> in a <commutative ring>, reduction has the displayed <ring isomorphism>. The kernel is their intersection, equal to their product. Choose $e_i\in\prod_{j\ne i}I_j$ with $e_i\equiv1\pmod{I_i}$; then $\sum_i a_ie_i$ lifts any prescribed residue tuple. The equality between intersection and product follows from $1=u+v$, $u\in I$, $v\in J$, which writes each $x\in I\cap J$ as $xu+xv\in IJ$.
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