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 indexed by an arbitrary index set, the product is defined by the set of all functions such that for all i which becomes an R-Module under pointwise operations, and "
================================================== =============================================
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 is not neccesarily the same as and so on.
However, if we take then
Then the product
BUT the set notation implies that the order or sequence of the product does not matter i,e that etc
Is this correct?
It worries me that, again considering the example of the operations in would be of the form
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 and . In particular, in the pointwise operations why choose subtraction instead of addition.
Can anyone clarify these matters for me?
Peter
Thank you johng
I am still struggling with the idea that, in your example is the same as (isomorphic to) .
My problem is that the elements of these products are ordered triples not just sets - so the elements of are of the form while the ordered triples of are of the form . As ordered triples these are not the same (of course, they are the same as sets of components - but I think the product of modules is dealing with elements that are ordered triples?
Can you comment.
I am also still struggling with the product as a set of mappings. Can you give some examples of the mappings in the case of A, B and C?
Peter