Lemma for Chinese Remainder Theorem

Question

I want to prove the following Lemma:

Let $R$ be a commutative ring with $1\neq 0$ and $I,J_{1}, J_{2},..., J_{n}$ be the ideals of $R$ such that $I+J_{i}=R$ $\forall 1\leq i\leq n$. Then, $I$, $J_{1}J_{2}...J_{n}$ are co-maximal.

I have no idea how to approach it.

Also, I want to know what extra information do we get if $J_{1}, J_{2},..., J_{n}$ are pairwise co-maximal i.e. $J_{i}+J_{j}=R$ $\forall 1\leq i\neq j \leq n$.

Answer 1

Hint:

The hypothesis means that for each $i=1,\dots,n$, there exists a pair $(x_i, y_i),\;x_i\in I,y_i\in J_i$, such that $x_i+y_i=1$.

You can show by induction on $n$ that the product $(x_1+y_1)\dotsm(x_n+y_n)$, which is equal to $1$, can be expanded as $X+Y$, where $X\in I,\: Y\in J_1\dotsm J_n$.