1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Determine all the ideals of the ring Z

  1. Mar 7, 2005 #1
    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. jcsd
  3. Mar 7, 2005 #2

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    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
  4. Mar 8, 2005 #3

    mathwonk

    User Avatar
    Science Advisor
    Homework Helper

    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.
     
  5. Mar 8, 2005 #4
    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 ?
     
  6. Mar 8, 2005 #5

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    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.
     
  7. Mar 8, 2005 #6

    mathwonk

    User Avatar
    Science Advisor
    Homework Helper

    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Determine all the ideals of the ring Z
  1. Ideals in Rings (Replies: 2)

Loading...