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?