I am seeking to understand Dummit and Foote's chapter on Representation Theory and Character Theory (see pages 840-843 attached).
On Page 840 D&F write: (see attachment)
Suppose we are given an FG-Module V.
We obtain an associated vector space over F and a representation of G as follows:
Since V is an FG-Module, it is an F-Module i.e. it is a vector space over F.
Also for each g G we obtain a map from V to V, denoted by , defined by
where is the given action of the ring element g on the element of V.
Since the elements of F commute with each it follows by the axioms of a module that for all and all we have:
that is, for each is a linear transformation.
My problem with the above text is
1. I do not see why we can assume that " the elements of F commute with each "
2 Hence I do not follow the assertion above that
Can someone please show me explicitly how this is the case?
Would appreciate some help?