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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?

velut luna
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1 Answers1

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(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.

oxeimon
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  • Thanks! Then in (3) can I consider, e.g., both $a_1$ and $a_2$ belong to, say, $N_1$? – velut luna May 24 '14 at 22:33
  • so the book is wrong to say that each $a_i \in N_i$? – velut luna May 24 '14 at 23:57
  • and in fact there can be more than $n$ terms in the sum? – velut luna May 24 '14 at 23:58
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    @user139981 I don't think oxeimon tackled that part of your question well. Yes, the book means to assert that each $a_i\in N_i$. There is no point in writing two separate things in $N_1$, for example, because the sum of the two is also in $N_1$, and you could just replace it by a single element of $N_1$. – rschwieb May 25 '14 at 12:08