I know that if we have a noncommutative ring the CRT (Chinese Remainder Theorem) doesn't work and I know that the CRT works for all PIDs (Principal Ideal Domains).
My question is:
In the case of the ring of Hurwitz quaternions, which is a noncommutative PID, will CRT work?
I need the CRT be valid in this way:
Let $R$ be a PID and $p_1, \dots, p_K$ elements in $R$. If the $p_k$ are pairwise coprime, and let $M=\displaystyle\prod_{k=1}^K p_k,$ and $M_k=M/p_k, \ k=1,\dots,K$ the function $$g(w_1,...,w_K)=(w_1M_1+\dots+w_KM_K)\bmod{MR}$$
is a one-to-one correspondence between $R/p_1R\times\dots\times R/p_KR$ and $R/MR$. Further, for any choice of $w_1\in R/p_1R,\dots, w_K\in R/p_KR$, $x=g(w_1,\dots,w_K)$ is the unique element of $R/MR$ with this property $$x \bmod p_1R=w_1, x \bmod p_2R=w_2,\dots, x \bmod p_KR=w_K.$$
Is this ok?!
I am confused because the Hurwitz ring is a PID (and an Euclidian Domain), so in my head it should work.