The ideal product is well-defined

In TAN, the product of two finitely generated ideals $(A)$ and $(B)$ was defined to be $(ab \ |\ a \in A, b \in B)$. Argue that this is a well-defined operator on the set of ideals of a fixed ring $R$.

More broadly, $IJ = (xy \ |\ x \in I, y \in J)$. If $I = (A)$ and $J = (B)$, then each $x$ is an $R$-linear combination of the $a_i \in A$, and each $y$ a combination of the $b_i \in B$. Since $R$ is a commutative ring, then, each $xy$ is an $R$-linear combination of $a_ib_j$ where $a_i \in A$ and $b_j \in B$; that is, $IJ \subseteq (ab \ |\ a \in A, b \in B)$. The reverse inclusion follows from this previous exercise.

A more modern treatment would derive the generating set characterization of $IJ$ from the more general definition.