Talk:Free module

message: What is a "free basis"? Shouldn't it just be "basis"?--129.70.14.127 23:58, 7 November 2007 (UTC)
replies: text: Done. --Beroal (talk) 15:35, 30 November 2010 (UTC)
thread_title:  Definition 

message: The article does not mention:
* projective modules, which under suitable restrictions are the locally free modules
* free ideal rings, and the commutative specialization principal ideal rings
* ideal class groups (measuring the difference between projective ideals and free ideals)
* grothendieck groups (measuring the same for f.g. modules)
* free resolutions, or the more popular finite free resolutions
* freeness is not preserved under Morita equivalence, the property generated by freeness under Morita equivalence is again the notion of projective module
* skew fields, the rings over which every module is free
The article's treatment of rank is poor.  Rank should be given its own section.
For the zero ring, or rings without identity, free modules are very poorly behaved.  For nonzero rings with identity: If E is a set, then every direct sum of E regular modules is determined up to isomorphism by the cardinality of E.  If E, F are two such sets, E is infinite, and E and F have distinct cardinalities, then R^(E) and R^(F) are not isomorphic.  Hence R^(E) has rank |E| if E is infinite.  For some rings, R^m and R^n can be isomorphic modules without m being equal to n.  This cannot happen for commutative rings, which should probably be mentioned.  (Rings where R^m = R^n implies m=n, have the IBN property (linked in the article), and m is called the rank of R^m.
For the zero ring, the definition of free basis is slightly incorrect as it confuses a set (with its elements indexed) with an "indexed set", more commonly called a function from E to M.  For nonzero rings, distinct e in E map to distinct elements of the module, but not for the zero ring (where all unital modules are free and isomorphic). JackSchmidt (talk) 19:40, 22 August 2008 (UTC)
replies: text: Hungerford has a reasonable discussion of free modules over rings without identity, FWIW. They are basically free (unital) modules over that universal unital ring containing the non-unital ring. JackSchmidt (talk) 19:21, 8 May 2009 (UTC)

text: I have written a small section that gives pointers to the relation with projective and flat modules, but I think that more properties you mention (i.e. projectives being "almost" locally free) shouldn't be on this page but on the page for projective modules (or on another, corresponding, specialised page). Konrad (talk) 06:02, 11 May 2012 (UTC)
thread_title:  Some areas for improvement 

message: show that Q is not a finitely generated Z-module  —Preceding unsigned comment added by 218.248.65.82 (talk) 06:35, 11 June 2009 (UTC)
thread_title:  module 

message: This claim in the introduction is linked to a source which says "vector space" will mean a module over the complex numbers. So, even if the general result is true, I think the link should be removed, and the result should be proven in the article.  — Preceding unsigned comment added by 98.248.228.230 (talk) 09:20, 27 November 2011 (UTC)
thread_title:  "Every vector space is free"