Chinese remainder theorem

Chinese remainder theorem: Let $R$ be a commutative ring with $1$ , and let $I_1,\dots,I_n\triangleleft R$ be ideals such that $I_i+I_j=R$ for all $i\neq j$ (pairwise comaximal, expressed via the sum of ideals). Then the natural map

R \longrightarrow \prod_{i=1}^n R/I_i,\qquad r\longmapsto (r\bmod I_1,\dots,r\bmod I_n)

is surjective with kernel $\bigcap_{i=1}^n I_i$ (the intersection of ideals), hence induces an isomorphism

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

of quotient rings. The right-hand side is the ring structure on the cartesian product given componentwise.