Is it true that if an ideal $I$ of ring $R$ can be denoted as the product of ideals $J$ and $K$ then $I \subseteq J$ and $I \subseteq K$?

Question

I just proof-read a proof of someone, and in the proof the assumption is used that if $I$ is an ideal of a ring $R$ such that $I = JK$ for some other ideals $J$ and $K$, then $I \subseteq J$ and $I \subseteq K$. The proof claims that this follows from the definition of the product of two ideals, but I cannot see why this should be true. Is it perhaps true under certain circumstances (in this case, for example, $I$ was a principal ideal, and $R$ was a Dedekind Domain)? Or is the assumption just wrong, and is the proof therefore plainly false? Or is it just true, and is it something I just don't see?

Answer 1

The ideal $JK$ is the additive subgroup generated by elements $ab$ with $a\in J$ and $b\in K$. All these $ab$ lie in $J$, since $a\in J$ and $b\in R$ and $J$ is an ideal. As $J$ is closed under addition, all elements of $JK$ lie in $J$.