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

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SUMMARY

The discussion centers on the question of whether the ideals of the direct product of rings R × S are necessarily of the form I × J, where I and J are ideals of R and S, respectively. The original poster initially believed this proposition to be true but later concluded that it is false. The discussion highlights the challenges in proving that any ideal of R × S can be expressed as A × B, where A is a subset of R and B is a subset of S, ultimately leading to the realization that the proposition does not hold.

PREREQUISITES
  • Understanding of ring theory and ideals
  • Familiarity with direct products of algebraic structures
  • Knowledge of the properties of ideals in rings
  • Basic proof techniques in abstract algebra
NEXT STEPS
  • Study the properties of ideals in direct products of rings
  • Learn about counterexamples in ring theory
  • Explore the structure of ideals in specific rings, such as integers and polynomial rings
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Mathematicians, particularly those specializing in abstract algebra, students studying ring theory, and anyone interested in the properties of ideals in algebraic structures.

fischer
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I want to answer this question:
Find all the ideals of the direct product of rings R \times S.
(I think this means show that the ideals are I \times J where I, J are ideals of R, S, respectively.)

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

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

can someone help me out?
thanks in advance.
 
Last edited:
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never mind. i got it.
the proposition is false...

here, i attached the solution.
 

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Hi,
Is this solution correct?
 

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