I am reading a book on Abstract and Linear Algebra and saw the following theorem:
"Suppose $M$ is an $R$-module, $T$ is an index set, and for each $t \in T$, $N_t$ is a submodule of $M$.
i) $\cap_{t\in T} N_t$ is a submodule of $M$.
ii) If {$N_t$} is a monotonic collection,$\cup_{t\in T} N_t$ is a submodule.
iii) $+_{t\in T}N_t$ = {all finite sums $a_1 + \cdots +a_m$: each $a_i$ belongs to some $N_t$} is a submodule. If $T$ = {$1, 2,.., n$}, then this submodule may be written as $N_1 + N_2 +\cdots +N_n = \{a_1 + a_2 + \cdots +a_n$ : each $a_i \in N_i\}$."
I have a few questions about (iii)
(1) Why the sum needs to be finite?
(2) Must different $a_i$ belong to $N_t$ with different $t$?
(3) Because it's finite sum, when the index set is infinite, the $a_i$s belong only to some $t$, but not all. Then why for finite $T$ = {$1, 2,.., n$}, the submodule must be of the form $\{a_1 + a_2 + \cdots +a_n$ : each $a_i \in N_i\}$. Can I consider sum of less than $n$ terms?
(1) Because you can't add up infinitely many elements of a module without some notion of topology and convergence.
(2) No.
(3) Yes you can consider sums of less than $n$ terms, though note that this is the same as letting some terms $a_i$ be 0.