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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:

  1. 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?

  2. "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.

Regards

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    Look up [invariant basis number](http://en.wikipedia.org/wiki/Invariant_basis_number).2011-05-15
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    Your first question may confuse "basis" with "minimal generating set". In a local ring they are the same thing, but in a general ring such as the integers Z, one can have large minimal generating sets such as { 1 }, { 2, 3 }, { 6, 10, 15 }, and { 30, 42, 70, 105 }. Your second question is definitely an IBN question.2011-05-15
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    Thank you for clarifying this. I know that the two notions are different but I appreciate that you stressed the difference2011-05-15
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    No problem. Then the first question is definitely misstated. Over every commutative (associative, unital) ring, every basis of any particular free module has the same cardinality. Local is irrelevant. All commutative rings have IBN.2011-05-15
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    @Jack Hi Jack well actually i just read the excerpt from my textbook again and you are right. It is about minimal generating sets. so what does "local" add here? What is the rational behind a module being local in general? Thanks in advance2011-05-15

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