I want to clear a confusion regarding the following paragraph from chapter 2 of Atiyah Macdonald.
Suppose that the ring $A$ is a direct product $\prod_{i=1}^{n}A_i$. Then the set of all elements of A of the form $(0, ... , 0, a_i, 0, ... , 0)$ with $a_i \in A_i$ is an ideal $a_i$ of $A$. The ring $A$, considered as an $A$-module, is the direct sum of the ideals $a_1, ... , a_n$.
This makes me think that while considering the direct sum of ideals as A-modules, they're trying to say - $(a_1, 0, ... , 0) \oplus (0, a_2, ... , 0) = (a_1, a_2, 0, ... , 0)$ whereas I think it should be that $(a_1, 0, ... , 0) \oplus (0, a_2, ... , 0) = ((a_1, 0, ... , 0), (0, a_2, ... , 0))$ by definition.
I know these two are equivalent upto isomorphism, but is identifying $(a_1, a_2, 0, ... , 0) \in A$ with $((a_1, 0, ... , 0), (0, a_2, ... , 0)) \in A_1 \oplus A_2$ a right way of interpreting it?