I know that assuming axiom of choice or equivalently Zorn's lemma , it can be proved that every non-trivial ring with unity has a maximal ideal (two sided ) . The wiki article on axiom of choice says that this statement regarding existence of maximal ideal in any non-trivial ring with unity is equivalent to axiom of choice , but I am not able to prove this converse implication . Please help . Thanks in advance .
The proof is not trivial to come by.
Originally given by Hodges, the proof shows a variant of Zorn's lemma can be proved from the assertion that every commutative ring with a unit has a maximal ideal.
Wilfrid Hodges, Krull implies Zorn, J. London Math. Soc. (2) 19 (1979), no. 2, 285--287.
Some decades later, Banaschewski gave a somewhat different proof of that same fact.
Bernhard Banaschewski, A new proof that “Krull implies Zorn”, Math. Logic Quart. 40 (1994), no. 4, 478--480.
Both paper are not long, and quite readable granted that you're comfortable with reading choice-related papers.
Marcel Erné, A primrose path from Krull to Zorn, Comment. Math. Univ. Carolin. 36 (1995), no. 1, 123–-126.
Is also related to this proof, although I haven't really read that one.