I know that it seems very loose as a title but I hope this post will be beneficial to all the forum members.
One thing I like about free modules is that they help one define maps directly as we do in a vector space by just defining the images of the elements of a base (if it exists).
My questions are:
I have read somewhere that two minimal generating sets for a free module do not necessarily have the same cardinality, except if the corresponding ring is local. Is that true? What is the intuition behind a ring being "local" then?
"A map (module map of course) from our free module to itself is bijective iff it is injective." In which general setting is that statement true?
I hope that post end up containing many examples and counterexamples that are certainly beneficial to beginners like myself.