Choosing notation for remainders in proof of chinese remainder theorem (CRT)

Question

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 $e_i = ([0],\cdots,[1],\cdots,[0])$, where [0] is a remainder class (class of all element that are zero when mod with $a_1$).

Answer 1

$(0,\cdots,1,\cdots,0)$ is an acceptable abuse of notation, but $([0],\cdots,[1],\cdots,[0])$ does not improve on it much, if the goal is to have very descriptive notation.

I would recommend $e_i=(0+a_1,\cdots,1+a_i,\cdots,0+a_n)$, faithfully using the coset notation in each position, if the goal is more descriptive notation.

Using a lowercase roman letter for ideals also looks a little hard to read (to me at least) and it might be worth switching to capital or fraktur letters.

And I hope I don't sound like I'm blowing the importance of notation out of proportion here. All of these are perfectly acceptable depending on your audience and how you present your work.