I am reading John Dauns book "Modules and Rings". I am having problems understanding the notation of Section 1-2 Direct Products and Sums (pages 5-6) - see attachment).
In section 1-2.1 Dauns writes:
================================================== ===========================================
"1-2.1 For any arbitrary family of modules $\displaystyle M_i, i \in I $ indexed by an arbitrary index set, the product $\displaystyle \Pi \{ M_i | i \in I \} \equiv \Pi M_i $ is defined by the set of all functions $\displaystyle \alpha , \beta : I \rightarrow \cup \{ M_i | i \in I \} $ such that $\displaystyle \alpha (i) \in M_i $ for all i which becomes an R-Module under pointwise operations, $\displaystyle ( \alpha - \beta) (i) = \alpha (i) - \beta (i) $ and $\displaystyle ( \alpha (i) r for r \in R $"
================================================== =============================================
Can anyone clarify this notation for me - preferably using a simple example.
One specific issue is the following:
I imagine (possibly incorrectly) that the order or sequence of modules matters - that is $\displaystyle M_1 \times M_2 \times M_3 $ is not neccesarily the same as $\displaystyle M_2 \times M_1 \times M_3 $ and so on.
However, if we take $\displaystyle I = \{ 1, 2, 3\} $ then $\displaystyle \{ M_i | i \in I \} = \Pi \{M_1, M_2, M_3 \} $
Then the product $\displaystyle \Pi \{ M_i | i \in I \} \equiv \Pi \{ M_1, M_2, M_3 \} $
BUT the set notation implies that the order or sequence of the product does not matter i,e that $\displaystyle \Pi \{ M_1, M_2, M_3 \} = M_1 \times M_2 \times M_3 = M_3 \times M_1 \times M_2 $ etc
Is this correct?
It worries me that, again considering the example of $\displaystyle M_ 1, M_2, M_3 $ the operations in $\displaystyle M_1 \times M_2 \times M_3 $ would be of the form
$\displaystyle (x_1, x_2, x_3 ) + (y_1, y_2, y_3) = (x_1 + y_1, x_2 + y_2, x_3 + y_3) $
$\displaystyle (x_1, x_2, x_3 ) a = (x_1a, x_2a, x_3a )$
and in these triples (certainly in things like curves in 3 space) the order, I think, would matter.
-------------------------------------------------------------------------------------------
Then, I have to say I do not really have a good understanding of or feeling for what follows regarding the functions $\displaystyle \alpha $ and $\displaystyle \beta $. In particular, in the pointwise operations why choose subtraction instead of addition.
Can anyone clarify these matters for me?
Peter