Definition of direct sum of modules? I have just started studying modules, and am trying to figure out the definition of the direct sum of modules but I'm having trouble since different sources seem to give different definitions, for example:
MIT says:
The direct sum of the Mλ is the subset of restricted vectors:
$\bigoplus$ $M_{λ}$ := {($m_{λ}$) | $m_{λ}$ = 0 for almost all λ} 
Wolfram MathWorld says:
The direct sum of modules A and B is the module
A $\bigoplus$ B={a$\oplus$b|a $\in$ A,b $\in$ B},   
where all algebraic operations are defined componentwise.
[What is $\oplus$ anyway?]
My lecture notes say:
Define the direct sum of modules as the set theoretical product with the
natural addition and multiplication by elements of A.

The only one that makes sense to me is the last one, but it doesn't seem to agree with the other two

For finitely many summands everything agrees. Then $A\oplus B$ is what you wrote, and this holds in the same way for finitely many summands. For infinitely many summands there are differences, and we need to distinguish between $\oplus M_i$ and $\prod M_i$. This is expalined in books on rings and modules. A good example is the free $\mathbb{Z}$-module $\oplus_i \mathbb{Z}$, whereas the module $\prod_i \mathbb{Z}$ is not free, hence not projective.
