Let $R$ be a ring and let $I_1, \dots, I_n \subset R$ be ideals, which are pairwise coprime. Let furthermore $\pi_j:R \to R/I_j,\; r \mapsto r + I_j$, $j=1, \dots, n$ be the projection to the $jth$ quotient ring.
I know that for all $j \in \{1, \dots, n\}$, the ideals $I_j$ and $\cap_{i \neq j}I_i$ are coprime. For all $j \in \{1, \dots, n\}$ we take $d_j \in I_j$ and $e_j \in \cap_{i \neq j}I_i$ s.t. $d_j + e_j=1$. $$\Rightarrow \pi_i(e_j)=\pi_i(1-d_j)$$ This is now equal to $0$, if $i \neq j$ and equal to $1+I_j$, if $ i=j$. I don't see why we get this.
Many thanks for any help!
Ideals $I_j$ and $\bigcap_{i\ne j} I_i$ are coprime, which means that $I_j + \bigcap_{i\ne j} I_i = R$. Thus for $1\in R$ there are $d_j\in I_j$ and $e_j\in \bigcap_{i\ne j} I_i$ with $d_j+e_j=1.$
Then $\pi_i(e_j) = 0$, since $e_j\in I_i$ for $i\ne j$.