I am trying to understand the notions of rank of an R-Module, free-module, basis, etc.(adsbygoogle = window.adsbygoogle || []).push({});

I would like to understand this line (expand on it, find some critical examples/counterexamples ,etc) that I am quoting from Dummit & Foote:

"If the ring R=F is a field, then any maximal set of F-linearly independent elements is a basis for M (the module) . For a general integral domain, however, an R-Module M of rank n need not have a basis, i.e., need not be a free R-Module even if M is torsion free, so some care is necessary...."

Before this came the definition: For an integral domain R, the rank of an R-module is the maximal number of R-linearly independent elements of M.

Thank you

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# Free-module, basis, rank

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