I have a question related to this: Projective modules
I'm trying to understand the "philosophy" of the statement, because it seems too similar to the statement "a module is free iff every element can be written uniquely as a finite linear combination of elements of a basis".
Is this "projective basis" property saying this:
a module P is projective iff every element in P can be written as a finite linear combination of some elements of P?
We lose uniqueness in the expression as a sum: in the elements of P, in the elements of R, and in the number of terms (so the concept of "rank" wouldn't make sense). Is this all, or am I misunderstanding the statement?
Any other intuition related to that property is also appreciated.