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).