This question may be really stupid and several posts related to my questions have been posted, but none of them addressed my confusion. In page 20 of Atiyah and Macdonald, if I am reading it correctly, they say something as follows.
Let $A$ be a commutative ring and if $A=\prod_{i=1}^{n}A_{i}$ is a direct product of rings $A_{i}$, then the set of all elements of $A$ of the form $$(0,\cdots, 0,a_{i},0,\cdots,0),$$ where $a_{i}\in A_{i}$ is an ideal $\mathfrak{a}_{i}$ of $A$.
I am confused by this definition. So the set is the set of all the elements $(a_{1},0,0,\cdots,0)$ with $a_{1}\in A_{1}$, $(0,a_{2},0,0,\cdots,0)$ with $a_{2}\in A_{2}$, and so on until $(0,0,\cdots,0,a_{n})$ with $a_{n}\in A_{n}$.
Then, why there is still a subscript for $\mathfrak{a}_{i}$? Why does this ideal still depend on the choice of $i$? I don't know if this question is obvious, but it is pretty confusing to me..
