Sum of intersection\intersection of sum of ideals

Question

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. $I_1 \cap (I_2+ I_3)= (I_1\cap I_2) + (I_1\cap I_3)$.

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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.)

Answer 1

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.