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"What does the correspondence theorem tell us about ideals in Z[x] that contain x[itex]^{2}[/itex] + 1?

My thinking is that since Z[x]/(x[itex]^{2}[/itex] + 1) is surjective map and its kernel is principle and generated by x[itex]^{2}[/itex] + 1 since x[itex]^{2}[/itex] + 1 is irreducible. This implies ideals that contain x[itex]^{2}[/itex] + 1 are principle and isomorphic to C.

I'm not sure if (a) my reasoning is right and (b) what answer this question is trying to get from us.

Thank you for your help

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# Correspondence theorem for rings

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