## An equivalent characterization of ideal products

In TAN, we defined the product of (finitely generated) ideals $I = (A)$ and $J = (B)$ to be $I \star J = (ab \ |\ a \in A, b \in B)$. We can also define an ideal product $IJ = \{\sum_T x_iy_i \ |\ T\ \mathrm{finite}, x_i \in I, y_i \in J\}$. Prove that $IJ = I \star J$.

We did this previously.