In TAN, the product of two finitely generated ideals and was defined to be . Argue that this is a well-defined operator on the set of ideals of a fixed ring .
More broadly, . If and , then each is an -linear combination of the , and each a combination of the . Since is a commutative ring, then, each is an -linear combination of where and ; that is, . The reverse inclusion follows from this previous exercise.
A more modern treatment would derive the generating set characterization of from the more general definition.