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Question

# Determine all the ideals of the ring $\mathbb{Z}[x] /\left(2, x^{3}+1\right)$.

Solution

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$\hspace*{5mm}$Let the $\Bbb{Z}[x]/(2,x^3+1)$ be the ring. By the Third Isomorphism Theorem we get

$\Bbb{Z}[x]/(2,x^3+1) \equiv (\Bbb{Z}[x] / (2)) / ((2,x^3+1) / (2)) \equiv \Bbb{Z}/2\Bbb{Z}[x]/(x^3+1)$

$\hspace*{5mm}$ $\Bbb{Z}/2\Bbb{Z}$ is a field, so $\Bbb{Z}/2\Bbb{Z}[x]$ is a principal ideal domain. By $\textbf{exercise 5}$ there is a one to one carrespondence between the ideals of $\Bbb{Z}/2\Bbb{Z}[x]/(x^3+1)$ and the divisors of $(x^3+1)$

$\hspace*{5mm}$ Therefore the ideals of $\Bbb{Z}[x]/(2,x^3+1)$ have the form $(2\mod (2,x^3+1),p(x)\mod (2,x^3+1))$ for $p(x)=1$, $p(x)=x+1$, $p(x)=x^2-x+1$, and $p(x)=x^3+1$.

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