TITLE: What is internal direct sum or internal direct product in Dummit and Foote?
QUESTION [5 upvotes]: I refer to Dummit and Foote Chapter 10.3 specifically pages 351,353,354,356 and 357.

Does Exercise 10.3.21 on pages 357 (By the way, there's some errata here. Condition (iii) should be $i_1,...,i_k$) define a notion of internal direct sum (of unital $R$-submodules of a unital $R$-module over a unital, but not necessarily commutative, ring $R$)?


I think this is an internal direct sum for an infinite or a finite index set that generalises the notion of internal direct sum for a finite index set given in page 354.


Do we have a notion of 'internal direct product'?


For the finite case, I believe this is the '$N_1 + ... + N_k$' part of Proposition 10.5 in page 353.

For the finite or infinite case, I believe this is the 'the (unital $R$-)submodule of $M$ generated by (the union of) all the $N_i$'s' part of Condition (i) of Exercise 10.3.21 because '$N_1 + ... + N_k$' in Proposition 10.5 is actually equal to (see page 351) the (unital $R$-) 'submodule of $M$ generated by (the union of) all the $N_i$'s' such that Condition (i) generalises the '(1)' in Proposition 10.5.

Therefore: I think of internal direct product of $N_i$'s of $M$ as $\sum_{i \in I} N_i = R\{\bigcup_{i \in I} N_i\}$, which like external direct product and external direct sum, is always defined. And then I think of internal direct sum as not always defined but, whenever defined, as equal to internal direct product.

Possibly relevant: 'Semidirect product'. This wikipedia page: https://en.wikipedia.org/wiki/Direct_sum_of_groups#Generalization_to_sums_over_infinite_sets


Context: I'm trying to understand the direct sum parts of graded rings and graded ideals in later in Chapter 11.5. I'm hoping these can be internal instead of just external. I ask more here.

Edit 1: Thank you for the upvotes or views. I feel like all the hours I spent trying to understand this seemingly minor thing was really worth it.
Edit 2: For (not necessarily Abelian) groups: Internal direct product/sum in groups: Is join and independent equivalent to unique expression?

REPLY [5 votes]: 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.