Product Of Ideals - Why is the sum necessary?

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Discussion Overview

The discussion revolves around the concept of the product of ideals in ring theory, specifically questioning the necessity of including finite sums of products of elements from two ideals. Participants explore examples and seek clarification on why the definition requires summation to maintain the properties of an ideal.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant expresses confusion about why the product of two ideals I and J cannot simply be defined as the set of elements of the form ij without including finite sums.
  • Another participant argues that defining the product without sums would not yield an ideal, as ideals must be closed under addition.
  • A participant seeks a specific example where the product defined without summation fails to be an ideal, indicating that they suspect such examples exist but are struggling to identify one.
  • Another participant suggests that matrix rings might provide a suitable context for finding such an example.
  • One participant claims that if either I or J is a principal ideal, then the product can be defined without sums, as it would still hold the necessary properties.

Areas of Agreement / Disagreement

Participants generally agree that the inclusion of finite sums is necessary for the product of ideals to maintain the structure of an ideal, though there is ongoing debate about specific examples and contexts where this necessity is illustrated.

Contextual Notes

Participants note that examples may vary depending on the type of ring considered, particularly suggesting that principal ideal domains may not provide the necessary complexity to illustrate the point.

jd102684
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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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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.
 
Last edited:
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!
 
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.
 
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 }​
 

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