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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?

  2. jcsd
  3. Apr 23, 2012 #2
    Try finding a counterexample.
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