  In this paper, we study conditions guaranteeing that a product of ideals
defines a Golod ring. We show that for a $3$-dimensional regular local ring (or
$3$-variable polynomial ring) $(R , \m)$, the ideal $I \m$ always defines a
Golod ring for any proper ideal $I \subset R$. We also show that non-Golod
products of ideals are ubiquitous; more precisely, we prove that for any proper
ideal with grade $\geq 4$, there exists an ideal $J \subseteq I$ such that $IJ$
is not Golod. We conclude by showing that if $I$ is any proper ideal in a
$3$-dimensional regular local ring and $\mfa \subseteq I$ a complete
intersection, then $\mfa I$ is Golod.
