Define a "ideals" as sub-object that are (but isomorphisms) kernel of some morphim. The orders of ideals is in order bijection with the orders of the extremal quotients, that are (but isomorphism) just the surjective morphisms. This bijection is merely:

 $I \mapsto (q_I: R \to R/I)$ and $q \mapsto Ker(q)$. 

Now given two ideals $I, J \subset R$ their "product" $I\ast J$ is the ideal generated by these two, or the minimal ideal containing both $I$ and $J$, in toher words it is $I \vee J$ in the order of the ideals of $R$.
Then $I\ast J = Ker (q_{I, J})$ where $q_{I, J}: R \to Q$ come from the pushout of $q_I: R\to R/I$, $q_J: R\to R/J$, because this pushout is $q_I\vee q_J$ in the  order of the extremal quotients.

Excuse my poor English.