Product of Ideals makes no sense to me

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In summary, the product of ideals I and J is a set defined as the sum of all possible products of elements from I and J, where n represents the number of elements being multiplied. In the case of 4Z*2Z inside Z, the index n is not enumerating the elements in a particular order, but rather represents the number of elements being multiplied in a product. This can result in different answers depending on the order in which the elements are multiplied. However, in general, the product of two ideals mZ and nZ is equal to mnZ. The correct definition of the product of ideals should include a universal quantifier to account for all possible products.
  • #1
Credulous
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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.
 
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  • #2
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.
 
  • #3
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: $$ m\mathbb{Z}*n\mathbb{Z}=mn\mathbb{Z}?$$
 
  • #4
Yes, that's right. In general the product is contained in the intersection of the ideals - since each aibi 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
 
  • #5
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. that's where the sums come in.
 

1. What is a product of ideals?

A product of ideals is a mathematical concept in abstract algebra that involves multiplying two ideals together to create a new ideal. This process is similar to multiplying two numbers together, but instead of numbers, we are working with algebraic structures called ideals.

2. How is a product of ideals different from a product of numbers?

The main difference between a product of ideals and a product of numbers is the type of structure we are working with. In a product of numbers, we are multiplying two elements from a number system, such as integers or real numbers. In a product of ideals, we are multiplying two subsets of a ring or algebraic structure.

3. Why does the product of ideals sometimes not make sense?

The product of ideals may not make sense in certain cases because it depends on the properties of the ring or algebraic structure we are working with. If the ring is not commutative, for example, the product of ideals may not be well-defined. Additionally, if the two ideals do not have any elements in common, their product may be the zero ideal, which can be confusing.

4. What is the purpose of studying products of ideals?

Studying products of ideals is important in abstract algebra because it helps us understand the structure and properties of rings and other algebraic structures. It also allows us to define new ideals and investigate their relationships with existing ideals. Furthermore, products of ideals have applications in fields such as number theory and algebraic geometry.

5. Can you give an example of a product of ideals?

One example of a product of ideals is in the ring of polynomials over a field. Let's say we have the ideals (x) and (x^2 + 1), which represent the set of polynomials with only x and the set of polynomials with only x^2 + 1, respectively. The product of these two ideals would be the ideal (x^3 + x), which contains all polynomials that can be expressed as the product of a polynomial with only x and a polynomial with only x^2 + 1.

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