[Math] Ideals and the Chinese Remainder Theorem

Question

It is very intuitive to think that since modular arithmetic partitions the integers into equivalence classes (or residue classes), the sentence

…the Chinese remainder theorem can be generalized to ideals.

is true.

Here is what I imagine: Say the residue classes mod 3 are

\begin{align}[0]&=\{…,-6,-3,0,3,6,…\}\\
[1]&=\{…,-5,-2,1,4,7,…\}\\
[2]&=\{…,-4,-1,2,5,8,…\}
\end{align}

Are the ideals (subsets) of integers generalized through the CRT mod $N= n_1\cdot n_2\cdot n_3= 3\cdot 4\cdot 5=60$ all the $x$ integers for any given $3$ integers $a_1,a_2,a_3$ that fulfill

\begin{align}
x\equiv a_1 \mod 3\\
x\equiv a_2 \mod4\\
x\equiv a_3 \mod 5
\end{align}

?

Answer 1

No. The generalised Chinese remainder theorem is an abstract version in the context of commutative rings, which states this:

Let $R$ be a commutative ring, $I_1,\dots, I_n$ pairwise relatively prime ideals (i.e. $I_k+I_\ell=R\;$ for any $k\ne \ell$). Then

1. $I_1\cap\dots\cap I_n=I_1\dotsm I_n$.
2. The canonical homomorphism: \begin{align} R&\longrightarrow R/I_1\times\dotsm \times R/I_n,\\ x&\longmapsto (x+I_1,\dots ,x+I_n), \end{align} induces an isomorphism: $$R/I_1\cap\dots\cap I_n=R/I_1\dotsm I_n\simeq R/I_1\times\dotsm \times R/I_n.$$