A Course in Finite Group Representation Theory
《A Course in Finite Group Representation Theory》 · 第 39 页 · Section 2.3「Ring Homomorphisms, Factor Rings」，Proposition 12
To end the present section we want to establish the so-called Chinese remainder theorem. Proposition 12. Let $R$ be a ring and $\mathfrak a_1,\ldots,\mathfrak a_n\subset R$ pairwise coprime ideals, i.e., assume that $\mathfrak a_i+\mathfrak a_j=R$ for $i\ne j$. Then, writing $\pi_i:R\to R/\mathfrak a_i$ for the canonical projections, the homomorphism $\varphi:R\to R/\mathfrak a_1\times\cdots\times R/\mathfrak a_n$, $x\mapsto(\pi_1(x),\ldots,\pi_n(x))$, is surjective and satisfies $\ker\varphi=\mathfrak a_1\cap\cdots\cap\mathfrak a_n$. In particular, it induces an isomorphism $R/(\bigcap_{i=1}^n\mathfrak a_i)\cong\prod_{i=1}^n R/\mathfrak a_i$, where $\prod_{i=1}^n R/\mathfrak a_i=R/\mathfrak a_1\times\cdots\times R/\mathfrak a_n$ is the ring-theoretic product of the residue class rings $R/\mathfrak a_i$.