It is known that Choice is equivalent to the statement "every nonzero ring has a maximal ideal".
But, if I'm reading this abstract correctly (I don't have access to the full paper), it is also equivalent to the more specific statement that only specifies commutative rings.
Is there a proof from "maximal ideals in commutative rings" to "maximal ideals in general rings", that doesn't pass though Zorn's lemma, Choice, etc., and stays entirely (or at least mostly) in the algebraic domain?
I thought about using the abelianization of the ring, which has a maximal ideal, but in general, maximal ideals don't pull back to maximal ideals, and to get the desired conclusion, this has to work for any ring.
Does anyone know of such an argument?