# Determine all the ideals of the ring Z

1. Mar 7, 2005

### b0mb0nika

determine all the ideals of the ring Z[x]/(2,x^3+1)

i'm a bit confused b/c this is a quotient ring.
would the ideals be all the polynomials which are multiples of x^3+1, with their free term an even number ?

2. Mar 7, 2005

### Hurkyl

Staff Emeritus
I'm not sure what you're saying, but it sounds wrong.

Here are a couple of ideas that might help:

(1) Recall that any ideal is the kernel of some homomorphism...

(2) You can lift any ideal of Z[x]/(2,x^3+1) to an ideal of Z[x]...

(3) Z[x]/(2,x^3+1) is a rather small ring...

Last edited: Mar 7, 2005
3. Mar 8, 2005

### mathwonk

the ideals of a quotient ring are those ideals of the original ring that contain the ideal in the bottom of the quotient, so you are looking for all ideals of Z[X] that contain both X^3 +1 and 2.

4. Mar 8, 2005

### b0mb0nika

i was thinking kind of the same thing as mathwonk.. but i'm not sure how to write those ideals..
if they contain X^3+1 and 2.. wouldn't they have to be multiples of them ?

5. Mar 8, 2005

### Hurkyl

Staff Emeritus
Does (4) contain 2?

Whatever method you try, I recommend also trying my third suggestion -- the ring is small, so you can explicitly write all of the elements of the ring, and directly work out all of its ideals.

6. Mar 8, 2005

### mathwonk

Hurkyl's suggestions are always valuable.

Also, remember the elements of the smallest ideal containing u and v, consists of all linear combinations of form au+bv, with a,b, in the ring.

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