Chinese remainder decomposition

For comaximal ideals, a quotient ring decomposes as a product of quotients.

Chinese remainder decomposition

Chinese remainder decomposition: Let $R$ be a commutative ring and let $I_1,\dots,I_n$ be ideals that are pairwise comaximal. Then the natural homomorphism $R\to \prod_{i=1}^n R/I_i$ induces an isomorphism

R\Big/\bigcap_{i=1}^n I_i \;\cong\; \prod_{i=1}^n R/I_i.

In particular, if $I$ and $J$ are comaximal, then $R/(I\cap J)\cong R/I\times R/J$ .

This is the standard Chinese Remainder Theorem formulated as an explicit isomorphism of quotient rings when ideals are comaximal (i.e., their sum is the whole ring). It relates the intersection of comaximal ideals to a product ring and produces idempotents yielding the splitting in idempotent decompositions .