I've known the Chinese Remainder Theorem like
Let the Ring $R$ and its Ideals $I$ and $J$ s.t. $I+J=R$
Then $R/(I \cap J) \simeq R/I +R/J$
The question is
First) What is the condition of $IJ =I\cap J$ statement is true?
I.e.) Want to know the case that $(R/IJ) \simeq (R/(I \cap J))\simeq (R/IJ)$ is true.
Second) Let the R is the commutative ring with the unity like a domain.
Then, Does $(R/IJ) \simeq (R/(I \cap J))\simeq (R/IJ) $ always true?
My proof(It is just my thought) : $I,J \subset I+J \subset R$ and $I+J =R$
Hence $1_R \in I+J$, I and J are relatively prime. Therefore $IJ = I \cap J$
(Might be a wrong, But I could't find which point I was wrong.)
Third) Find the integers $m,n$
$Z[i]/\langle 4+8i \rangle \simeq Z_m \times Z_n$
Is it possible utilizing the chinese remainder theorem with this question?
e.g.) $(Z[i]/\langle 4+8i \rangle) \simeq (Z[i]/\langle 4 \rangle) \times (Z[i]/\langle 1+2i \rangle) $
Any help or advice would be appreciated.
Thank you.
Best Answer
I am not sure of necessary conditions for your first question, but a sufficient condition for $IJ = I \cap J$ is that $I,J$ are comaximal (ie. $I + J = R$), the same condition in the chinese remainder theorem.
If you are asking if the chinese remainder theorem is true in an integral domain (which is what it looks like you are asking in the second question), then the answer is no. Consider $\mathbb{Z}$ and the ideals $I = 4\mathbb{Z}$ and $J = 2\mathbb{Z}$. Then, $\mathbb{Z}/IJ \cong \mathbb{Z}/8\mathbb{Z}$ and $\mathbb{Z}/(I \cap J) \cong \mathbb{Z}/4\mathbb{Z}$. Hence, $\mathbb{Z}/IJ \not \cong \mathbb{Z}/(I\cap J)$.
I think the answer to the third question is Yes. This is because $4 + 8i = 4(1 +2i)$. Defining $I = \langle 4 \rangle $ and $J = \langle 1 + 2i \rangle $ we see that $\langle 4 + 8i\rangle = IJ$. Moreoever (and you should check this), $I + J = \mathbb{Z}[i]$. Thus, by the chinese remainder theorem $\mathbb{Z}[i]/ \langle 4+8i \rangle \cong \mathbb{Z}[i]/\langle 4 \rangle \times \mathbb{Z}[i]/ \langle 2 + i \rangle$ which as you elude to is $\mathbb{Z}_4[i] \times \mathbb{Z}_5$