I am reading Dummit and Foote Ch 10: Modules and in particulat section 10.3 Generation of Modules, Direct Sums and Free Modules.
I am having problems understanding and interpreting Theorm 6 - see attached pdf for the two relevant pages of D&F.
Problems I have are as follows:
1. We are told to "identify A as a subset of F(A) by a where is the function which is 1 at a and zero elsehwhere"
Thus F(A) - but how can A be a subset of F(A)? The members of F(A) are functions , , , ... etc. The entries are functions not just simple points or elements of A. The nature of the elements of A and F(A) seem different, so can we "identify" the elements of A as a subset of F(A)?
Further to this point where is the function a used as the proof progresses from this point?
2. Following the quote above regarding , we read:
"We can, in this way, think of F(A) as all finite R-linear combinations of elements of A by identifying each function f with the sum + + ..... + where f takes the value at and is zero at all other elements of A. Moreover, each element of F(A) has a unique expression as such a formal sum."
So you can see what assumptions I am working on, I have sketched my understanding of functions f, g F(A) and also my understanding of - see pdf attached called sketches of functions
Given that D&F describe f as taking on the value at and being zero at all other elements, how can we show that each element of F(A) has a unique expression as a formal sum of the form + + ..... + .
[I am also somewhat confused by the fact that the functional values of f are , , etc and not , , etc ]
Can anyone help clarify this theorem for me?