Question Let $A$ be a commutative ring and $I_1, I_2, \cdots, I_n$ be ideals of $A$. Any two of the above ideals are relatively prime, i.e., $$I_i+I_j=A, \forall i\ne j.$$ Are $I_i$ and $\bigcap\limits_{j\ne i}$ relatively prime?
This question comes up when I consider the example $\mathbb{Z}$. Suppose that $I_1=2\mathbb{Z}$, $I_2=3\mathbb{Z}$ and $I_3=5\mathbb{Z}$, any two of which are relatively prime. If we intersect $I_2$ and $I_3$, we get $I_2\cap I_3=15\mathbb{Z}$, also relatively prime with $I_1=2\mathbb{Z}$.
As $\mathbb{Z}$ is a typical commutative ring, I guess this is probably true with any commutative ring. But how to prove this? How to use the definition of reletively prime stated above?
Any help would be highly appreciated! Thank you in advance!