general chinese remainder theorem
1. Proposition
Let \(R\) be a ring and \(\mathfrak{a}_1,...,\mathfrak{a}_n\) be coprime Ideals. Then
\begin{align*} R/(\bigcap \mathfrak{a}_i) \cong R/(\prod_{i=1}^n \mathfrak{a}_i) \cong \otimes_{i =1}^n R/\mathfrak{a}_i \end{align*}2. Proof
2.1.
2.2.
consider the ring-homomorphisms
\begin{align*} \varphi: R \rightarrow& \otimes_{i=1}^n R/\mathfrak{a}_i \\ x \mapsto& (\pi_1(x),...,\pi_n(x)) \end{align*}