Product Of Ideals - Why is the sum necessary?

In summary: This means that if you want to define the product of two ideals, you have to define it as an operation on the individual ideals. This is what the book does.
  • #1
jd102684
4
0
So I'm learning about Ring Theory and have gotten to Ideals. My book tells me that the product of two ideals I and J or a ring R (standard * and + operations) is the set of all finite sums of elements of the form ij where i is in I and j is in J. I'm having trouble coming up with an example as to why we can't just define the product of I and J to be just the elements of the form ij (not any finite sum).

The definition without the sum seems to work for the rings I've tried (R being the integers Z, and ideals being nZ and mZ for different m and n, for example). I suspect it breaks down with matrices, or maybe polynomials, but I'm having trouble nailing down a specific example. Guidance would be much appreciated. Thanks!

- JD
 
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  • #2
jd102684 said:
I'm having trouble coming up with an example as to why we can't just define the product of I and J to be just the elements of the form ij (not any finite sum).

You could do that, but it wouldn't be all that useful because that set isn't an ideal. An ideal has to be closed under addition.

IJ is the smallest ideal containing all the elements of the form ij, and this forces finite sums of elements to be included as well.
 
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  • #3
Right, I'm sorry, I guess I didn't exactly state my question. I'm trying to see an example where the way I defined IJ (without the sum) is NOT an ideal. I can't seem to work out an example where the summation is necessary to maintain closure, though I know there's probably plenty basic examples out there!
 
  • #4
jd102684 said:
Right, I'm sorry, I guess I didn't exactly state my question. I'm trying to see an example where the way I defined IJ (without the sum) is NOT an ideal. I can't seem to work out an example where the summation is necessary to maintain closure, though I know there's probably plenty basic examples out there!

Gotcha. I'll try to come up with an example. I think you may be right that you need to consider e.g. matrix rings.
 
  • #5
You don't have to leave the nice world of commutative algebra.

However, what you do need to do is leave the world of principal ideal domains. I claim the following:

If I or J is principal, then IJ = { ij | i in I and j in J }​
 

1. Why is the sum of ideals necessary in product of ideals?

The sum of ideals is necessary in product of ideals because it allows us to define a new ideal that contains all the elements that can be obtained by multiplying elements from each of the original ideals. In other words, the sum of ideals allows us to capture the entire set of possible products between elements of the two ideals.

2. Can we use other operations besides sum in product of ideals?

No, the sum is the only operation that can be used in product of ideals. This is because the sum is the only operation that preserves the properties of an ideal, such as closure under addition and absorption of scalar multiplication. Other operations may not necessarily result in an ideal when applied to two ideals.

3. How does the sum of ideals relate to the product of rings?

The sum of ideals is similar to the product of rings in that both operations involve combining elements from two different sets. However, the product of rings combines elements from two rings, while the sum of ideals combines elements from two ideals. Additionally, the product of rings results in a new ring, while the sum of ideals results in a new ideal.

4. Is the sum of ideals commutative?

Yes, the sum of ideals is commutative. This means that the order in which we add the ideals does not matter and the result will be the same. This can be easily proven using the properties of the sum of ideals, such as closure under addition and associativity.

5. Can the sum of ideals be empty?

Yes, the sum of ideals can be empty if one or both of the original ideals are empty. This is because the sum of ideals is defined as the set of all possible sums of elements from the two ideals, and if there are no elements in one or both ideals, then there are no possible sums. However, in most cases, the sum of ideals will not be empty as it will contain at least the zero element.

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