Theorem (CRT) : Let $a_1,\cdots ,a_r$ be ideals in a ring $R$, such that $a_i + a_j = R$, for all $i \ne j$. Let $\pi_i: R \mapsto R/a_i$ denote the surjection. Then the map $$\eta : R \mapsto R/a_1 \times \cdots R/a_r$$
given by $\eta(x) = (\pi_1(x_1),\cdots \pi_r(x_r))$ is onto.
Proof : A typical element in $R/a_1 \times \cdots R/a_r$ is of the form $(\pi_1(x_1),\cdots \pi_r(x_r)) = \pi_1(x_1)e_1 + \cdots +\pi_r(x_r)e_r$, where $e_i = (0,\cdots,1,\cdots,0)$ one is at $i th$ postion.
Here I have a doubt should I write $e_i = (0,\cdots,1,\cdots,0)$ or $e_i = (,\cdots,,\cdots,)$, where  is a remainder class (class of all element that are zero when mod with $a_1$).