Product of Ideals makes no sense to me

Credulous

The definition of the Product of ideals I and J is $$IJ = \{a_1b_1+a_2b_2 + ...+ a_nb_n | a_i \in I, b_j \in J, n \in \mathbb{N}\}$$.

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.

Office_Shredder

The n isn't enumerating the elements of I and J in a particular order, it's just saying you can have any finite sum. For example for 4Z*2Z: every sum using one element from each of 4Z and 2Z is allowed. So 4*2, 16*6, -12*4 are all in 4Z*2. Every sum using two elements from each is allowed: 4*2+32*6, 4*4+40*14 are in 4Z*2Z. Etc.

Credulous

So in essence, that would mean any combination of both sets, they would all contain factors of 8. So that would mean $$4\mathbb{Z}*2\mathbb{Z}=8\mathbb{Z}?$$or more generally: $$ m\mathbb{Z}*n\mathbb{Z}=mn\mathbb{Z}?$$

Office_Shredder

Yes, that's right. In general the product is contained in the intersection of the ideals - since each a_ib_i is contained in the inersection, adding them up is also contained in the intersection, hence each element of I*J is contained in the intersection of I*J, but as we can see from this example you can get smaller ideals than the intersection by multiplying

mathwonk

its not your fault, there is a missing universal quantifier in that definition, it should say for ALL n.

basically take all products xy where x is in one ideal and y is in the other, and then take the smallest ideal containing all those products. thats where the sums come in.