Is the Transitivity of Ideals Always Guaranteed in Rings?

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Discussion Overview

The discussion revolves around the transitivity of ideals in rings, specifically whether an ideal of an ideal is necessarily an ideal of the larger ring. Participants explore examples, counterexamples, and the conditions under which transitivity may or may not hold.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant questions whether an ideal I of an ideal J can fail to be an ideal of the ring R, seeking conditions for guaranteed transitivity.
  • Another participant provides an example using the integers, suggesting that if I is an ideal of J and J is an ideal of R, then I should also be an ideal of R, contingent on certain conditions being met.
  • A different participant presents a counterexample involving the ideals I = ⟨x²⟩ and J = ⟨x⟩ in the ring R = ℝ[x], arguing that I is not an ideal of R despite being an ideal of J.
  • One participant critiques the notion of discussing "an ideal of an ideal," suggesting that proper ideals are not rings and thus complicate the discussion.
  • Another participant notes that some authors may refer to "rngs," implying that ideals of ideals could be valid in that context, although this usage is not universally accepted.
  • Errors in the earlier example are pointed out, specifically regarding the lack of a unit in J affecting the ideal generated by x².
  • A participant acknowledges the error but raises a point about the potential closure under addition, questioning whether irrational multiples of x² could still pose issues.

Areas of Agreement / Disagreement

Participants express differing views on the transitivity of ideals, with some providing examples that support their positions while others offer counterexamples. The discussion remains unresolved, with no consensus on the conditions that guarantee transitivity.

Contextual Notes

Some participants reference specific properties of ideals and rings, such as the necessity of closure under operations and the implications of lacking a unit. The discussion highlights the complexity of definitions and assumptions in the context of ideals.

esisk
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Hi All,
If I is an ideal of J and J is an ideal of R (the ring)... Is it possible that I is not an ideal of R. When are we guaranteed to have this transitivity, if at all? Thank you
 
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I'm still trying to work out the details, but would this be an example?

Let Z be the ring of integers, then 2Z is an ideal of Z and 4Z is an ideal of 2Z. But 4Z is also an ideal of Z. I would guess that is you have the situation you described, then I would always be an ideal of R.

Let R be a ring and let J be an ideal of R. We must first ensure that J is a ring in order for it to have any ideals. So as long as its closed under subtraction and multiplication, its a ring, which we can confirm. Next let I be an ideal of J. Then all elements in I must be in R, since I is an ideal of J and J is one of R. So, as long as we can take any element in R, multiply by an element in I and still stay in I, then I would say that I is an ideal of R. Give that a try and see what happens.
 
esisk said:
Hi All,
If I is an ideal of J and J is an ideal of R (the ring)... Is it possible that I is not an ideal of R. When are we guaranteed to have this transitivity, if at all? Thank you

In general this is not true: Consider I= \langle x^2 \rangle = \{ fx^2 : f\in J= \langle x \rangle \} with both I and J considered as subsets of R= \mathbb{R} [x]. By definition I ideal of J and J ideal of R, but I is not an ideal of R since ax^2 \notin I for all a\neq 0,1 such that a\in \mathbb{R} . As for conditions that guarantee transitivity I can't think of any but that J=R.
 
Well, in all honesty, it's a bit silly to talk about "an ideal of an ideal" since we define (particularly in commutative algebra) ideals to be subsets of rings and proper ideals are not rings (ie. they are not subrings).
 
Some authors use the word "ring" to refer to rngs, so ideals of ideals makes sense (because any ideal of a rng is a rng).

In what I've read I've never seen that usage -- although most authors will make an explicit statement they are not adopting that convention.

I'm going to assume the opening poster is talking about rngs.
 
Jose, there is an error in your example: because J doesn't have a unit, the ideal of J generated by x2 is not equal to x2J.
 
Hurkyl said:
Jose, there is an error in your example: because J doesn't have a unit, the ideal of J generated by x2 is not equal to x2J.

You're right, but couldn't we "close" it under addition and then it still won't have any, for example, irrational miltiples of x^2?
 
I believe that is true.
 

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