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Question

Determine all the ideals of the ring Z[x]/(2,x3+1)\mathbb{Z}[x] /\left(2, x^{3}+1\right).

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\hspace*{5mm}Let the Z[x]/(2,x3+1)\Bbb{Z}[x]/(2,x^3+1) be the ring. By the Third Isomorphism Theorem we get

Z[x]/(2,x3+1)(Z[x]/(2))/((2,x3+1)/(2))Z/2Z[x]/(x3+1)\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} Z/2Z\Bbb{Z}/2\Bbb{Z} is a field, so Z/2Z[x]\Bbb{Z}/2\Bbb{Z}[x] is a principal ideal domain. By exercise 5\textbf{exercise 5} there is a one to one carrespondence between the ideals of Z/2Z[x]/(x3+1)\Bbb{Z}/2\Bbb{Z}[x]/(x^3+1) and the divisors of (x3+1)(x^3+1)

\hspace*{5mm} Therefore the ideals of Z[x]/(2,x3+1)\Bbb{Z}[x]/(2,x^3+1) have the form (2mod(2,x3+1),p(x)mod(2,x3+1))(2\mod (2,x^3+1),p(x)\mod (2,x^3+1)) for p(x)=1p(x)=1, p(x)=x+1p(x)=x+1, p(x)=x2x+1p(x)=x^2-x+1, and p(x)=x3+1p(x)=x^3+1.

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