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.