# If $(a)$ and $(b)$ are principal ideals of $R$, when is $(a)(b) = (ab)$?

Assume $R$ is a ring and $(a)$ and $(b)$ are principal ideals in $R$. Under what conditions is the product of the ideals equal to the principal ideal generated by the product, i.e. when is $(a)(b)=(ab)$?

My guess is that they're guaranteed to be equal if $R$ is a U.F.D. (or maybe just an integral domain?), but I do not know how to prove this for an arbitrary ring. I can't think of any counterexamples that would suggest my intuition to be wrong though. Any thoughts?

• Are you assuming your ring is commutative? Commented Feb 27, 2018 at 21:34
• I suppose. The example that I'm trying to apply it to is $(x^3-2x) = (x)(x^2-2) \subseteq R = \mathbb{Q}[x]$. I just want to know when we can break $(ab)$ up as $(a)(b)$, since it seems to be a pretty useful fact. Commented Feb 27, 2018 at 21:45

For any commutative ring, it is elementary to show $(a)(b)=(ab)$.
$ab\in (a)(b)$, so $(ab)\subseteq (a)(b)$.
Everything in $(a)(b)$ is of the form $\sum_{i=1}^nar_ibs_i=ab(\sum_{i=1}^nr_is_i)\in (ab)$, so $(a)(b)\subseteq (ab)$.
• Ah I see where that matches up with my proof for the example I'm working on. All we need is commutativity to guarantee $(a)(b) \subseteq (ab)$. Commented Feb 27, 2018 at 21:48