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!

Proving an ideal in R.

  1. Apr 23, 2012 #1
    1. The problem statement, all variables and given/known data
    Let I and J be ideals in R. Is the set K = {ab|a is an element of I, b is an element of J} an ideal in R?

    2. Relevant equations
    Conditions for an ideal, I of a ring R;
    (i)I is nonempty,
    (ii)for any c,e ε I: c-eεI
    (iii)for any c ε I, rεR: rc, cr ε I.

    3. The attempt at a solution

    Let a,bεK.

    K is not empty since it contains 0.
    K seems to satisfy condition (iii) since r(ab)=(ra)b, and raεI since I is an ideal. Then (ra)bεK.
    Also, (ab)r=a(br), and since brεJ, a(br)εK.

    But nothing jumps out at me when I examine ab - cd , where a,cεI and b,dεJ.

    Is it possible this is not an ideal?

    Thanks.
     
  2. jcsd
  3. Apr 23, 2012 #2

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

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




Similar Discussions: Proving an ideal in R.
  1. Maximal Ideal of R (Replies: 3)

  2. Ideals, elements r^n (Replies: 6)

Loading...