Kaplansky's theorem and Axiom of choice - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T21:47:21Z http://mathoverflow.net/feeds/question/107819 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/107819/kaplanskys-theorem-and-axiom-of-choice Kaplansky's theorem and Axiom of choice AliReza Olfati 2012-09-22T06:55:35Z 2012-09-22T07:13:42Z <p>Kaplansky in his paper titled by <a href="http://www.jstor.org/discover/10.2307/1970252?uid=3738280&amp;uid=2129&amp;uid=2&amp;uid=70&amp;uid=4&amp;sid=21101201021821" rel="nofollow"><strong>Projective Modules</strong></a> gave an important and essential theorem as follow:</p> <p><strong>Theorem</strong>: Let \$R\$ be a ring, \$M\$ an \$R\$-module which is a direct sum of (any number of) countably generated \$R\$-modules. Then any direct summand of \$M\$ is likewise a direct sum of countably generated \$R\$-modules.</p> <p>But if you could take a look to the pattern of his proof, he applied the well ordering Theorem for proving it.</p> <p>I am thinking about the relation of his proof with the <strong>well ordering theorem</strong>. More precisely I am thinking about the answer of the following question:</p> <blockquote> <p>Question: Is the Kaplansky theorem equivalent with the Axiom of Choice or it can be proved with the weeker axiom(i.e.boolean prime ideal theorem)?</p> </blockquote>