Modules, Direct Sums, Confusion I have a couple of very closely related questions to direct sums of modules. The texts I am currently using are 


*

*Commutative Algebra with a View Toward Algebraic Geometry - David Eisenbud, 

*Introduction to Commutative Algebra - Atiyah & Macdonald, 

*Linear Algebra Done Right - Axler.



Question 1:

Both the Eisenbud and Atiyah & Macdonald texts define the direct sum of modules as "If $M$ and $N$ are $A$-modules, then the direct sum of $M$, $N$ is the module $M \oplus N = \{ (m,n) : m \in M, n \in N\}.$ They also give the structure, but that's not really important for my question.
Axler instead defines the sum of two subspaces (his text is for vector spaces, but since the vector spaces are necessarily modules I feel they should be consistent) as $U_1 + U_2 = \{ u_1 + u_2 : u_1 \in U_1, u_2 \in U_2\},$ for $U_1, U_2$ subspaces of a vector spave $V$. Then says the sum is direct if each element of $V$ can be written uniquely as a sum $u_1 + u_2$. 
He gives the example
"If $$U = \{ (x,y,0)\in \mathbb{F}^3 : x,y \in \mathbb{F} \}$$ and $$W = \{ (0,0,z) \in \mathbb{F}^3 : z \in \mathbb{F} \}$$ then $$U \oplus W = \mathbb{F}^3.$$
However using the previous definition from Eisenbud and Atiyah & Mac I would have written $$U \oplus W = \{ \left( (x,y,o),(0,0,z) \right) : (x,y,0) \in U, (0,0,z) \in W\}$$ which could very well likely be mapped isomorphically to $\mathbb{F}^3$?

Question 2:

Axler also states $V = U \oplus W$ if and only if $V = U+W$ and $U \cap W = \{0\}$. Is this true in general for modules? Something Eisenbud writes appears to be using this fact. Specifically, in his justification for when a short exact sequence of modules splits.

Question 3:

Wikipedia says that the direct sum of two modules is the smallest module containing both. Does it really contain both? It seems to me using the definitions of Eisenbud and Atiyah & Mac that the "summands" are only embedded into the direct sum. Not actually contained in. I.e, if we have $M \oplus N$, it contains $M$ in the sense that it contains $\{ (m,0) : m \in M\}$.

Question 4

In Eisenbud, when justifying when a short exact sequence splits, he says the following. Let $$0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$$ be a short exact sequence. If we have a map $h: C \to B$ such that $g \circ h = ID_C$, then the short exact sequence splits and $B \cong A \oplus C$. $ \textit{ Reason: }$ $\mathrm{Im}(h)$ and $\mathrm{Im}(f)$ are disjoint, but together they generated $B$.  So $B = \mathrm{Im}(f) \oplus \mathrm{Im}(h)$.
Is he using the proposition from Axler I mentioned above? It is strange to me to define direct sum the way Eisenbud did, in a way that seem like "how to make new modules from old modules" but then say something like this. Of course, he may be expecting more background than I have. 
I see why the images are disjoint (except, they share the 0 element), and I think I see why they "generate" B, but I am not entirely sure on that. It seems to me that every element of $B$ is either going to be hit by the injection coming in from $A$, or the injection coming in from $C$ (through the map $h$), but how do I deal with elements not in either image? Once I understand how we get this isomorphism, I see how to carry the sum of images to the actual objects.

Question 5:

From Eisenbud: "These maps are enough to identify a direct a direct sum: That is $M$ is a $\textbf{direct summand}$  of a module $P$ iff there are homomorphisms $a: M \to P$ and $b: P \to M$ whose composition $ba = ID_M$, then $P \cong M \oplus \mathrm{Ker}(b)$."
Can someone provide a proof here? I think I proved it using the splitting lemma, but the text gives this fact before the splitting lemma and I dont want to be circular. I also want to prove it bare bones to avoid "waving a magic wand" over it by applying a theorem, I need to understand the mechanics here to understand this well enough to move on.

Whats going on with all this direct sum stuff? Can someone please help
  me straighten this out?

 A: To address questions 1-3. In a way, there are two concepts of a direct sum, and some books actually make a clear distinction between internal direct sums and external direct sums.
If you have two submodules of an "ambient" module, $M,N\subseteq W$, then you can form their sum as a new submodule $M+N=\{w=m+n\mid m\in M,n\in N\}\subseteq W$. And if each element of the sum, $w\in M+N$, can be represented uniquely in this way, then we say that this sum is a direct sum, and thus we have an internal direct sum $M\oplus N$ of the submodules $M$ and $N$. From what you wrote above, this seems to be Axler's definition. From this point of view, a statement such as $U\oplus W=\mathbb{F}^3$ makes perfect sense: elements of $U$ and $V$ are actually elements of the same module, so they can be added together and compared with the entire $\mathbb{F}^3$.
The construction used in the other two books, where $M\oplus N=\{(m,n)\mid m\in M,n\in N\}$, is that of an external direct sum $M\oplus N$ of the modules $M$ and $N$. Note that here we don't care where $M$ and $N$ come from. For all we know, they can (and often do) consist of elements of different "nature". This construction does create a new kind of elements $(m,n)$ which are neither in $M$ nor in $N$.
Now, for two submodules $M,N\subseteq W$, assuming $M\cap N=\{0\}$, we can create both their internal and their external direct sum. But they are isomorphic, so people tend to abuse the language and the notation a little bit and not distinguish them from each other. Or in other words, it's usually clear from the context which type of direct sum has been constructed or is being discussed, so authors don't care to make the distinction.
Question 4. Pick an arbitrary $b\in B$, and let $x=b-h(g(b))$. Then
$$g(x)=g[b-h(g(b))]=g(b)-[\underbrace{g\circ h}_{\operatorname{Id}_C}\circ g](b)=g(b)-g(b)=0,$$
i.e. $x\in\operatorname{Ker}(g)=\operatorname{Im}(f)$, and
$$b=x+h(g(b))\in\operatorname{Im}(f)+\operatorname{Im}(h).$$
That's why they generate all of $B$.
A similar trick should work for Question 5 too.
