Equivalent Forms of AC - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T17:21:43Z http://mathoverflow.net/feeds/question/118199 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/118199/equivalent-forms-of-ac Equivalent Forms of AC Cheet 2013-01-06T13:02:55Z 2013-01-07T11:50:28Z <p>There are many algebraic equivalences of AC in the literature. A famous one is "every ring with identity has a maximal ideal". </p> <p>Where can I find this equivalences, specially those in rings theory !? Can you name some others ? Or a good references would be okay. </p> http://mathoverflow.net/questions/118199/equivalent-forms-of-ac/118200#118200 Answer by Asaf Karagila for Equivalent Forms of AC Asaf Karagila 2013-01-06T13:21:05Z 2013-01-06T13:43:29Z <p>One good source would be Rubin &amp; Rubin <strong>Equivalents of the Axiom of Choice</strong>, where there is a section devoted for algebraic forms. There are two editions to this book, the second one is from the 1980's and contains a lot more information (naturally).</p> <p>The book is a bit out dated and you can try Howard &amp; Rubin's <strong>Consequences of the Axiom of Choice</strong> (which also has a <a href="http://consequences.emich.edu/CONSEQ.HTM" rel="nofollow"><strong>website</strong></a>), there you could find principles which are (usually) independent of ZF, but are provable from ZFC; you can check those against the axiom of choice (Form 1) and see whether or not they imply it, but this is not done automatically.</p> <p>Another good place to check would be Herrlich's <strong>The Axiom of Choice</strong> which contains several equivalences (and "disasters") from several mathematical fields, in particular algebra (although not too much of it).</p> http://mathoverflow.net/questions/118199/equivalent-forms-of-ac/118270#118270 Answer by M Carl for Equivalent Forms of AC M Carl 2013-01-07T11:50:28Z 2013-01-07T11:50:28Z <p>You might also like Thomas Jech's book "The axiom of choice", where equivalences are discussed and some mathematics without choice is demonstrated.</p>