I'm having trouble wrapping my head around notions of free modules and here's why (quoting from the book):
So let M be a unital R-module (R commutative with identity). Suppose also that M is finitely generated, say . M is said to be freely generated by if
Furthermore, M is called free if it is freely generated by a finite set.
I'm confused as to what assumptions are being made all over the place, so here are my questions (I'll always assume I'm talking about unital modules over commutative rings with 1)
1) The way the notion of freely generated is presented makes it seem that if I say "M is an R-module freely generated by " then I'm implicitly assuming it is finitely generated. Is this true? Can it happen that M is not finitely generated but freely generated? If not then:
2) If M is freely generated by , can it happen that M is generated by more or less elements? If not then:
3) This condition looks suspiciously familiar to linear independence in vector spaces, but terms like "linear dependence" and "basis" are never used; the book even says that free modules behave very much like vector spaces. Is it wrong to think of this as the analogous of linear independent sets in vector spaces?
Any clarification is very welcome!
EDIT: the reason I'm asking is because right after there is a theorem stating M is freely generated by if and only if M can be uniquely expressed by finite sums of . The proof is short and vague so I was trying to fill in the details but that brings me to question 1) above. Is this theorem just stating 1 result (namely that the representation of elements is unique) or 2 (that there IS a representation of the elements AND furthermore it's unique)? Or rather: is the fact that by definition of freely generated or a consequence of it?