Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question
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

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 $A$.

share|cite|improve this answer

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

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.