Relationship between Ideals and subgroups ??

Question

I am just wondering if Product of two Ideals is again an Ideal.

Let $I$ and $J$ be ideal of Ring, then is $IJ=\{ij: i\in I, j\in J\}$ still an ideal?

Answer 1

No, $\{\,ij\mid i\in I, j\in J\,\}$ generates an ideal, which the smallest ideal that contains these product, and can be described as the set of all finite sums of such products.

In case of principal ideals, generated by $a$ and $b$, the product is again a principal ideal, generated by $ab$.