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There are many algebraic equivalences of AC in the literature. A famous one is "every ring with identity has a maximal ideal".

Where can I find this equivalences, specially those in rings theory !? Can you name some others ? Or a good references would be okay.

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up vote 5 down vote accepted

One good source would be Rubin & Rubin Equivalents of the Axiom of Choice, 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).

The book is a bit out dated and you can try Howard & Rubin's Consequences of the Axiom of Choice (which also has a website), 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.

Another good place to check would be Herrlich's The Axiom of Choice which contains several equivalences (and "disasters") from several mathematical fields, in particular algebra (although not too much of it).

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@Asaf: Thanks a lot – user30428 Jan 6 '13 at 15:44

You might also like Thomas Jech's book "The axiom of choice", where equivalences are discussed and some mathematics without choice is demonstrated.

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I don't know how many algebraic equivalents of AC appear in Jech's book. And I've been through it a lot. I agree it's a useful reference, but I don't think it fits this question very well. – Asaf Karagila Jan 7 '13 at 12:09

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