What is internal direct sum or internal direct product in Dummit and Foote?
Let me know if this answers your question. Let's let $R$ denote a unital ring. There are two similar but slightly distinct notions of internal and external direct sum, which I think is at the core of the question here. First and foremost, let $M$ denote an $R-$module, and let $N_1,\ldots, N_k$ denote submodules of $M$. In particular, as sets $N_i\subseteq M$ for each $i$. We say that $M$ is an internal direct sum of the $N_i$, denoted by $$ M=\bigoplus_{i=1}^k N_i$$ if every element of $M$ can be written uniquely as a sum of elements in the $N_i$. That is, for every $m\in M$, there exists a unique tuple $(n_1,\ldots, n_k)$ such that $m=\sum n_i$. It is equivalent to require that $N_1+\cdots+N_k=M$ and $N_i\cap N_j=\varnothing$ for $i\ne j$.
There is a slightly different notion of (external) direct sum where we take a collection of $R-$modules $N_1',\ldots, N_r'$ and we say that $M$ is an (external) direct sum of the $N_i'$ if there exists an isomorphism $\phi:M\to \bigoplus_{i=1}^k N_i'$. I.e. $$ \boxed{M\cong \bigoplus_{i=1}^k N_i'}$$ There is a bit of a distinction here, because we need to define this operation $\oplus$ for modules that do not both belong to a bigger module a priori. This is defined by the familiar rule $$ A\oplus B=\{(a,b): a\in A,b\in B\}$$ subject to the obvious $R-$module structure. So, being an external direct sum can be translated into the terminology of the internal direct sum as follows: $M$ is the external direct sum of $\{N_i'\}_{i=1}^k$ $$ \phi:M\xrightarrow{\sim} \bigoplus_{i=1}^k N_i'$$ if and only if there exist $N_i\subseteq M$ with $\phi(N_i)=N_i'$ for $i=1,\ldots, k$ and in fact $M$ is the internal direct sum of the $N_i$. That is, the $N_i'$ define $N_i=\phi^{-1}(N_i')$ so that $M$ is an internal direct sum of the $N_i$.
example: We should interpret what $\mathbb{R}^2=\Bbb{R}\oplus \Bbb{R}$ means. It means that there is a pair of subspaces $L_1,L_2$ of $\mathbb{R}^2$, each isomorphic to $\mathbb{R}$ so that $\mathbb{R}^2$ is their direct sum. In particular we can take $L_1$ to be the $x-$axis and $L_2$ to be the $y-$axis. These choices are far from unique.
Anyway, as you might know: this carries over almost verbatim to the case of an infinite indexing set, except that for $I$ a general indexing set, $\bigoplus_{i\in I}N_i$ consists of the finite sums of elements in the various $N_i$. So, you can re-define these notions in that case as an exercise.
If you are really interested in direct products, i.e. $M=\prod_{i=1}^k N_i$, then you should notice that for $R-$modules, finite products are isomorphic to finite coproducts (direct sums). I.e. $$ \prod_{i=1}^k N_i\cong \bigoplus_{i=1}^k N_i$$ and so the discussion carries over verbatim. In the case of infinite products, we get distinct notions: $$ \prod_{i\in I} N_i\not\cong \bigoplus_{i\in I} N_i$$ but you can still define the analogous notion of "internal" direct product using the same strategy.
- Yes, it does.
- Well, the factors of a product are clearly submodules of the product, but the issue is that since addition is finitary, it can never be additively generate the whole product.
I have never seen the notion of an "internal direct product" entertained, but there could be something to be said about characterizing it.
Proposition 10.5 proves that for finite sets, the direct sum and direct product coincide.
If it is helpful, here is my version of explaining how internal/external sums are related. Maybe it will help you see why there is a finitary constraint on sums, and not on products.