Let $R$ be a non-trivial commutative unital ring and let $I,J$ be ideals of $R$. The ideal product is defined as $IJ:=\{ i_1j_1+\dots+i_lj_l:l\ge 1, i_n\in I, j_n \in J, n\in\mathbb{N} \}$. Prove that $IJ$ is an ideal of $R$.
My approach was to prove that $IJ=I\cap J$, which implies that $IJ$ is an ideal.
Let $i\in I\cap J$, then $i\in I, J$, $=i_kj_k$ for some $i_k, j_k\in J$. But $i_kj_k\in IJ$, thus $i\in IJ$. Thus $I\cap J \subset IJ$.
For the other direction, I know that it is true, so I will skip the proof thereof. However, I'm not so sure that my proof above is correct since then $P^2=PP=P\cap P = P$, which is not true in general.
Can someone please point out what my error could possibly be? Maybe I didn't understand the definition of the product of ideal the way it was to be meant?
Best Answer
The problem with your proof is that $i\in I\cap J$ doesn't mean necessarily that $i=i_kj_k$ for some $i_k\in I$, $j_k\in J$. To see that this is not always true we can take the counterexample given by Bob Jones. Indeed, if $I=J=(2)$, then if your idea were right, we could write $2=(2a)(2b)$ for $2a, 2b\in (2)$, and this would imply that $4\mid 2$, which is clearly false.
In general the result you want to prove is only true if $I$ and $J$ are comaximal ideals, i.e., $I+J=R$. In general, to prove that $IJ$ is an ideal of $R$ we need to show:
i) For every $a,b\in IJ$, $a+b\in IJ$. To show this let's write $a=\sum_{i=1}^n x_iy_i$ and $b=\sum_{j=1}^m x_j'y_j'$, then $$a+b=\sum_{k=1}^{n+m}z_kw_k,$$ where $z_k=x_i$ and $w_k=y_i$ for $1\le k\le n$, and $z_k=x_j'$ and $w_k=y_j'$ for $n+1\le k\le n+m$. As $z_k\in I$ and $w_k\in J$ we have that $a+b\in IJ$.
ii) For every $r\in R$ and $a\in IJ$, $ra\in IJ$. Again, let's write $a=\sum_{i=1}^n x_iy_i$, so $$ra=r(\sum_{i=1}^n x_iy_i)=\sum_{i=1}^n(rx_i)y_i.$$
Since $rx_i\in I$ and $y_i\in J$ we deduce that $ra\in IJ$. Hence, $IJ$ is an ideal of $R$.