 Chinese remainder theorem for ideals

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 $e_i\equiv 1(\mathrm{mod}{I}_i)$; then ${\sum }_i{a}_i{e}_i$ 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 $xu+xv\in IJ$.