Find ideals of ring I am stuck with a homework problem.
Let $R=\mathbb{Z}[\sqrt{ -3}]$.
a) Find an ideal $I$ of $R$ such that $(4) \subsetneq I \subsetneq R$. Explain why the inclusions $\subsetneq$ in my example are strict.
b) Now find another ideal $J$ of $R$ such that $(4) \subsetneq J \subsetneq R$ again explain why the $\subsetneq$ are strict and explain why $J \neq I$.
c) Do there exist ideals $I_1, I_2$ of $R$ such that $(4) \subsetneq I_1 \subsetneq I_2 \subsetneq R$? Justify your answer.
Inclusion is not strict because $(4)$ can not be equal to ideal $I$ and to $R$, but I can not find those ideals and have no idea how to do part c).
 A: Remember, for any ideal $I \trianglelefteq R$ the ideals above $I$ corresponds to the ideals of $R/I$. Thus it suffices to consider $R/(4)$ in your case.
Since $R \cong \mathbb{Z}[X]/(X^2+3)$ we have $R/(4) \cong \mathbb{Z}[X]/(X^2+3,4) \cong (\mathbb{Z}/(4))[X]/(X^2+3)$. Since $X^2+3 = (X+1)(X-1)$ over $\mathbb{Z}/(4)$ we can conclude by the Chinese remainder theorem that $R/(4) \cong \mathbb{Z}/(4) \times \mathbb{Z}/(4)$. Now it should be really easy to calculate the ideals.
A: For part (a)  we have $(4)\subsetneq (2)\subsetneq R.$  See if you can understand that and explain why the containments are strict.  Then play around with this basic example and the fact that $\sqrt{-3}$ is in your ring to come up with an example for (b).
A: (Very) short answer:
In the ring $R=\mathbb{Z}[\sqrt{-3}]$ we have unique prime factiorization. Therefore all ideals in $R$ are principal Ideals. Thus, your Ideal $I$ must be of the form $\langle a\rangle$ where $a$ is a proper factor of $4$. Due to the famous formula $(a+b)(a-b)=a^2-b^2$ we get
$$ (1+\sqrt{-3})(1-\sqrt{-3})=1-(-3)=4 $$
In fact, this is the prime factorization of 4 in $R$. Perhaps, you should prove this fact. Now you get $I=\langle 1+\sqrt{-3}\rangle$ and $J=\langle 1-\sqrt{-3}\rangle$ and this is the only possible choice up to permutation.
