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.

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