What does the maximal ideal of rings means? Though I know the meaning of ideal, I just couldnot figure out What does the maximal ideal of rings means? If possible please explain me the meaning of quotient ring as well.
A quotient ring is just like all the other quotient structures in algebra. Given a two-sided ideal we can partition into equivalence classes where the equivalence relation is and we usually denote by the coset notation . We can then define a ring structure on by and .
To show that's well-defined isn't obvious, and for anything more you'll need to see a book. I suggest Dummit and Foote or Herstein.
That is to say, assuming the Axiom of Choice, maximal ideals always exist. If you do not assume it, they may not.