This is an exercise from Ideals, Varieties and Algorithms by Cox, etc.
If $I_i$ and $J_i=\cap_{j\ne i}I_j$ are comaximal for all $i$, where $I_1,...,I_r$ are ideals in $k[x_1,...,x_n]$, prove that $I_1^m\cap I_2^m \cap \dots \cap I_r^m=I_1^m\cdots I_r^m$.
Definition: Two ideals $I$ and $J$ in $k[x_1,...,x_n]$ are said to be comaximal if and only if $I+J=k[x_1,...,x_n]$.
Some background: I have proved that if $I$ and $J$ are comaximal, then $I^s$ and $J^r$ are comaximal for all positive integers $s$ and $r$.
I have also proved that if $I$ and $J$ are comaximal, then $IJ=I\cap J$. I highly suspect that this fact should be used to prove the statement, but I couldn't figure out how to use it in my prove.
My attempt so far: (Sketch of my attempt, since it is too long) It suffices to show that $$I_1^m\cap I_2^m \cap \dots \cap I_r^m\subset I_1^m\cdots I_r^m$$
Let $f\in I_1^m\cap I_2^m \cap \dots \cap I_r^m$. Let $I_j=\left\langle f_{j1},...,f_{ji_j}\right\rangle$. Then $f$ can be expressed in any $I_j^m$ as sum of terms with multi-index $m$ on the generators. I will denote an expression of $f$ in $I_j^m$ by $EF_j$.
First we write
$$f=EF_1$$
Since $I_1^m$ and $(I_2\cap\cdots \cap I_r)^m$ is comaximal, $1$ is in their sum. So $1$ can be written as sum of polynomial in $I_1^m$ and polynomial in $(I_2\cap\cdots \cap I_r)^m$, denoted by $E_1$ and $EC_1$, respectively. Notice that $EC_1$ can be expressed in any of $I_2^m, ..., I_r^m$. We write it in $I_2^m$ and denote it by $EC_{12}$.
Now we multiply $f$ by $1$, so that we can prove $f$ is in the product of the $I_j^m$'s. This gives (using the simplified notation) $$f=f\cdot 1 = EF_1(E_1+EC_{12})=EF_1E_1+EF_1 EC_{12}$$
We see easily the second term is in $I_1^mI_2^m$. The first term can be transformed into $EF_2E_1$ so it is in $I_1^mI_2^m$, too.
Repeating this process is very cumbersome. When the sum gets larger, I cannot find the pattern and cannot figure out whether it could be extended to the next one.
My question is: Is there an easier way to do it? I am very sorry for the bad notations and long post. I would really appreciate any input!