Ideals of direct product of rings are direct product of respective ideals?

In summary, the conversation is about finding all the ideals of the direct product of rings R \times S, which can be shown to be of the form I \times J where I and J are ideals of R and S, respectively. The problem is that the speaker is unsure how to prove that any ideal of R \times S is of this form, and they have made attempts to show that (a, m), (b, n) \in K if and only if (a, n), (b, m) \in K, but without success. They ask for help, but then realize that the proposition is actually false. The speaker has attached a solution to the problem.
  • #1
fischer
4
0
I want to answer this question:
Find all the ideals of the direct product of rings [tex]R \times S[/tex].
(I think this means show that the ideals are [tex]I \times J[/tex] where [tex]I, J[/tex] are ideals of [tex]R, S[/tex], respectively.)

I think the problem is that I don't know how to show that any ideal of [tex]R \times S[/tex] is of the form [tex]A \times B[/tex], where [tex]A \subset R, B \subset S[/tex]. Showing that each are ideals should follow easily enough.

So I made attemps to prove that [tex](a, m), (b, n) \in K[/tex] iff [tex](a, n), (b, m) \in K[/tex] (where [tex]K[/tex] is an ideal of [tex]R \times S[/tex]), without success...

can someone help me out?
thanks in advance.
 
Last edited:
Physics news on Phys.org
  • #2
never mind. i got it.
the proposition is false...

here, i attached the solution.
 

Attachments

  • essay_on_quiz.pdf
    35.6 KB · Views: 479
  • #3
Hi,
Is this solution correct?
 

1. What are "ideals" in the context of direct product of rings?

Ideals are subsets of a ring that satisfy certain properties, such as being closed under addition and multiplication by elements of the ring. They can be thought of as generalizations of the concept of divisibility in integers.

2. How are ideals related to direct products of rings?

In the context of direct product of rings, ideals are subsets of the product ring that satisfy similar properties as those in individual rings. This means that the direct product of ideals is also an ideal in the product ring.

3. Can the ideals of a direct product of rings be expressed as a direct product of the respective ideals?

Yes, the ideals of a direct product of rings can be expressed as a direct product of the respective ideals. This is known as the ideal correspondence theorem.

4. What is the significance of the ideal correspondence theorem?

The ideal correspondence theorem allows us to understand the structure of the ideals in a direct product of rings. It also helps us to prove certain properties of ideals in a more systematic and efficient manner.

5. Are there any exceptions to the ideal correspondence theorem?

Yes, there are certain cases where the ideal correspondence theorem does not hold. For example, in non-commutative rings, the direct product of two ideals may not necessarily be an ideal in the product ring. Additionally, the ideal correspondence theorem may not hold for infinite direct products of rings.

Similar threads

  • Linear and Abstract Algebra
Replies
5
Views
839
  • Linear and Abstract Algebra
Replies
1
Views
769
  • Linear and Abstract Algebra
Replies
17
Views
4K
  • Linear and Abstract Algebra
Replies
19
Views
2K
  • Linear and Abstract Algebra
2
Replies
55
Views
3K
  • Linear and Abstract Algebra
Replies
3
Views
4K
Replies
3
Views
2K
  • Linear and Abstract Algebra
Replies
3
Views
1K
  • Linear and Abstract Algebra
Replies
1
Views
1K
Replies
14
Views
1K
Back
Top