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
Best Answer
Let $A,B$ be $R$-modules. The direct sum $A\oplus B= \{(a,b) | a\in A, b\in B \}$ is a module under component wise operations: $(a_1,b_1)+(a_2,b_2)=(a_1+a_2,b_1+b_2)$ and $r(a,b)=(ra,rb)$.
This extends to a direct sum of finitely many $R$-modules. However, for a direct sum of infinitely many $R$-modules, there is a further requirement that elements have all but finitely many components equal to $0$.