The definition of the Product of ideals I and J is $$IJ = \{a_1b_1+a_2b_2 + ...+ a_nb_n | a_i \in I, b_j \in J, n \in \mathbb{N}\}$$.(adsbygoogle = window.adsbygoogle || []).push({});

But for say 4Z*2Z inside Z, how is the index "n" defined? I just don't get it. If you have $$2\mathbb{Z} = \{0, 2, -2,...\}$$ and $$4\mathbb{Z} = \{0, 4, -4, ...\}$$ that means $$4\mathbb{Z}*2\mathbb{Z} = \{0, 8, 16, 48, 80, ... \}$$ which is all well and good, but what if you defined 2Z = {0, -2, 2, -4, 4, ...}, what's stopping you? And then you get a completely different answer: $$4\mathbb{Z}*2\mathbb{Z} = \{0, -8, -16, ...\}$$ which is completely different from the first! Of course you could confuse things by further scrambling the order... but you get the point.

My question is: at least for the integers, what determines the "n" in the sum, because this current notation is seriously confusing me.

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# Product of Ideals makes no sense to me

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