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How to determine all the ideals of the ring $\mathbb{Z}[X]/(2,X^3+1)$?

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Where are you stuck? –  Aryabhata Jan 27 '11 at 15:55
    
This is exercise 7 in section 9.2 of Dummit and Foote. Is it your homework? –  Baudrillard Jan 27 '11 at 16:01
    
@Moron: I am trying the find the linear combination of 2 and x^3+1. Because there is a one-to-one correspondence between every ideal of Z[X]/(2,X^3+1) and every ideal of Z[x] containing the ideal (2,x^3+1). –  user6308 Jan 27 '11 at 16:02
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2 Answers

The ring $A={\Bbb Z}[X]/(2,X^3+1)$ is isomorphic to ${\Bbb F}_2[X]/(X^3+1)$ where ${\Bbb F}_2$ is the field with 2 elements. The ideals of the quotient $R/I$ are in one-to-one correspondence with the ideals in $R$ containing $I$. The correspondence is given by the quotient map.

Thus, it's enough to decompose $X^3+1$ in irreducibles over ${\Bbb F}_2$. It is straightforward to check that the decomposition is $X^3+1=(X+1)(X^2+X+1)$. The two factors generate the only two non-trivial ideals in $A$.

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HINT: Use the fact that $\mathbb{Z}[x]/(2,x^{3}+1) \cong \mathbb{Z}_{2}[x]/(x^{3}+1)$

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