Direct Sums of Modules If $A$ is a ring and the $a_i$ (some of) its ideals then what do the following mean :
1) $A = a_1 \oplus a_2 \oplus \cdots \oplus a_n$ (How to interpret this equality)
2) $A/b_i$ where $A$ is given as before and $b_i = \oplus_{j\neq i} a_j$ (How to interpret $A/b_i$)
(the sums here should be seen as direct sums of modules)
The difficulty for me is that by definition of direct sum of modules,
$a_1 \oplus a_2 \oplus \cdots \oplus a_n$ is the set of all $n$-tuples $(x_1,x_2,\cdots,x_n)$ such that $x_i \in a_i$ for all $i$. Clearly, this cannot 
$\textit{equal}$ $A$. However, there exists a simple module isomorphism between the two. As the text (Atiyah McDonald, Commutative Algebra) does not make this clear, I am unsure if this is the correct interpretation.
For the second question, should $A/b_i$ be thought of as a quotient module or a quotient ring? If it is a quotient module then $b_i$ has to be a subset
of $A$, however this cannot be true. So, I am imagining that $A/b_i$ should be 
interpreted as $a_1 \oplus a_2 \oplus \cdots \oplus a_n/c_i$ where $c_i = a_1 \oplus a_2 \oplus \cdots a_{i-1} \oplus 0 \oplus a_{i+1} \cdots a_n$. I am not at all sure if $A/b_i$ can be treated as a quotient ring.
Thanks for your help.
 A: As to the direct sum of modules, you correctly define the outer version. If you then have a module $M$, and submodules $M_{i}$ such that $M = \sum M_{i}$ and $M_{i} \cap \sum_{j\ne i} M_{j} = \{ 0 \}$ for all $i$, then $M$ is called the inner direct sum of the $M_{i}$, and as you note $M$ is then isomorphic to the outer direct sum. So it is not uncommon to use the same symbol for both versions.
If a ring $A$ is the (inner) direct sum of ideals $a_{i}$, then the direct sum of modules $A = a_1 \oplus a_2 \oplus \cdots \oplus a_n$ is really a ring one too. This is simply because if $i \ne j$, then
$$
a_{i} a _{j} \subseteq a_{i} \cap a_{j} = \{ 0 \},
$$
so that products, as well as sums, are done componentwise.
A: As I said in my comment, if that is your definition of direct sum (called, outer direct sum), the equal sign in 1) is abuse of notation, it should be $\cong$. Sometimes 1) means that every element of $A$ can be written uniquely as a sum of elements of the various ideals.
Let's see what $A/b_i$ means, for $b_i = \oplus_{j\neq i} a_j$. Let's fix an $i$, say $i_0$. Suppose all $a_i$'s are different. From 1), we can write $a \in A$ as $a=\sum^{n}_{i=0} a_i$. Then $A/b_{i_0}$ consists of all the elements $a \in A$ in which the summand in the expression  $a=\sum^{n}_{i=0} a_i$ coming from $a_{i_0}$ is non-zero. [I.e. the quotient $A/b_{i_0}$ makes everything $0$, except those elements that, when written as a sum $a=a_0+a_1+...+a_{i_0}+...+a_n$, have $a_{i_0}\neq 0$.] 
If there exists a $j \neq i_0$ with $a_j=a_{i_0}$, then $A/b_i$ annihilates everything. 
P.S. I believe that in 2) you meant $b_i = \oplus_{j\neq i} a_j$ (note the $j$ as subscript of $a$).
