The Chinese Remainder Theorem

We prove the Chinese Remainder Theorem (CRT) for commutative rings and discuss how to compute with it. We also apply the Chinese Remainder Theorem to prove that every ideal in $\O_K$ is generated by two elements and determine the structure of $\mathfrak{p}^n/\mathfrak{p}^{n+1}$, where $\mathfrak{p}$ is a nonzero prime ideal of $\O_K$.