Chinese Remainder Theorem

Let \(R\) be a commutative ring and \(I_i\) pairwise coprime ideals for \(i = 1,...,j\) . Then for \(r_i \in R\) for \(i = 1,...,j\) there exists an \(r \in R\) such that

\begin{align*} r \equiv r_j \mod{I_j} \end{align*}

proof by induction