TITLE: A confusion about Axiom of Choice and existence of maximal ideals.
QUESTION [11 upvotes]: The proof of the theorem of statement that every ring has a maximal ideal uses zorn's lemma or the axiom of choice.Now, the defintion of ring as well as the definition of maximal ideal don't depend on the axiom of choice.Is that true? So, given a ring I should be able to check whether it has maximal ideal or not without his axiom of choice. Am I right? So, how can the truth of this statement depend on Axiom of Choice? What is wrong with my assumptions?
To make my question more concrete, assuming that axiom of choice is false, can any one give me a ring which has no maximal ideal? How can a maximal ideal (assuming AoC is correct) turn into a non-maximal ideal just by assuming that AoC is false ?
PS. By rings I mean rings with identity.

REPLY [7 votes]: How can a maximal ideal (assuming AoC is correct) turn into a non-maximal ideal just by assuming that AoC is false ?

Changing your set of axioms does not change sets (nor does it leave them the same, for that matter.)  It only makes sense to discuss the properties of sets once we have fixed axioms.  Your question is analogous to the question "how can the pair $\{x,y\}$ disappear just by assuming that the Axiom of Pairing is false?" which also would make no sense. In both cases you have an axiom telling you that $\exists S\,P(S)$ is true where $P$ is some logical property of sets, and if you didn't have the axiom, then you couldn't prove the existence of such an $S$.
The issue with the Axiom of Choice is admittedly more subtle because it postulates the existence of sets without also giving a concrete description of those sets in the way that the Axiom of Pairing does.  Indeed, it is impossible to give such a concrete description when you are dealing with infinite sets.
There is no general way to consider infinite sets as having an existence that is independent of one's choice of axioms for set theory.  We are finite beings so we cannot just "write down" the set and "check" whether it has a given property.  When we say something like "AC implies that the ring $R$ has a maximal ideal," it would be a mistake to believe that the ring $R$—and the collection of all ideals of $R$—already exist as concrete objects accessible to our senses, independently of our choice of axioms.  If this were the case then indeed it would just be a matter of "checking", but it isn't the case.