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I'm stuck on a problem proposed by my teacher, that is to prove if $A$ is a commutative ring with unity, for any ideals $I_1, I_2, I_3$ of $A$ the following are equivalent:

  1. $I_1+(I_2\cap I_3)= (I_1+I_2) \cap (I_1+I_3)$;
  2. $I_1 \cap (I_2+ I_3)= (I_1\cap I_2) + (I_1\cap I_3)$.

I've tried very hard so far, but nothing works, and I'm beggining to think there's a hypothesis missing. Any hints? (I'm supposed to use only basic facts about rings and modules, tensor products, and localizations.)

José Victor Gomes
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1 Answers1

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I do not have enough reputation to comment so typing it as an answer.

Yes, you are definitely missing assumptions for the second part. This is true if only if $I_2\subseteq I_1$ or $I_3\subseteq I_1$.

You may also want to look at this When does the modular law apply to ideals in a commutative ring.

Insignificant
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  • But remais the question: doesn't (1) replace the hypothesis $I_2 \subset I_1$ or something like that? I want to prove that the two statements are equivalent, not prove them separately. – José Victor Gomes Mar 03 '20 at 20:51
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    Ok, the question you pointed enlightened me: rings that satisfy the condition in my problem are called arithmetic rings. Still, I don't know how to prove it, haha. – José Victor Gomes Mar 03 '20 at 21:04