I have an exercise where I need to manipulate different sets of ideals. So suppose you have $I$ and $J$ two ideals, then what would $IJ$ be equal to? Would it be $$IJ =I \times J = \{(i,j)| i \in I, j \in J \} $$ Or maybe $$IJ= \{ij| i \in I, j \in J \}$$
Or something else?
The ideal $IJ$ is defined to be the set:
$$\left\{\sum_{i=1}^n r_is_i : r_i \in I, s_i \in J, n \in \mathbb{N}\right\}$$
The reason this is defined in this way, is because the set $\{ij \mid i \in I, j \in J\}$ is NOT an ideal (in particular, it is not necessarily an additive subgroup of the ring you are working in), since you can't tell that $i_1j_1 + i_2j_2 \in IJ$ For an easy example, see this post: Understanding the ideal $IJ$ in $R$
For ideals, it is the ideal generated by all the products $ij$, where $i\in I,\,j\in J$, in other words, it the set of all finite sums $$\bigl\{i_1j_1+\dots i_nj_n\quad\text{for some }n\in\mathbf N, i_k\in I, j_k\in J\}.$$ Note that if $I$ and $J$ are principal ideals, generated by $a$ and $b$ respectively, $IJ$ is but the principal ideal generated by $ab$.
