Minimal generating set of a free module over local ring - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T04:53:52Z http://mathoverflow.net/feeds/question/83544 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/83544/minimal-generating-set-of-a-free-module-over-local-ring Minimal generating set of a free module over local ring Ivo Jan 2011-12-15T18:33:37Z 2011-12-15T18:58:04Z <p>Greetings, in my studies I went into a statement "minimal generating set of a free module over a local ring is a free basis". The statement came without a proof, just with a reference to Kaplansky's theorem. I was unsuccessful trying to prove the statement myself, and I couldn't find the proof elsewhere either. I would be grateful for any hint.</p> http://mathoverflow.net/questions/83544/minimal-generating-set-of-a-free-module-over-local-ring/83548#83548 Answer by Francesco Polizzi for Minimal generating set of a free module over local ring Francesco Polizzi 2011-12-15T18:58:04Z 2011-12-15T18:58:04Z <p>Any free module \$M\$ is projective. Then, when \$M\$ is finitely generated, you can use the argument given in <a href="http://books.google.it/books?id=yJwNrABugDEC&amp;printsec=frontcover&amp;dq=matsumura+commutative+ring+theory&amp;hl=en&amp;sa=X&amp;ei=zEHqTtrqLcrS4QSFtdTsCA&amp;redir_esc=y#v=onepage&amp;q=kaplansky&amp;f=false" rel="nofollow">Matsumura, Commutative Ring Theory</a>, proof of Theorem 2.5 page 10.</p> <p>Matsumura also expains the argument used by Kaplanski ["Projective modules", Ann. Math. <strong>68</strong> (1958)] in the general case.</p>