Proving the Ideal Status of K in R: I and J as Ideals in a Ring R

[Rfields]

Homework Statement

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?

Homework 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.

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.

 

[micromass]

Try finding a counterexample.