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About LinkBacksWhat 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 maximal ideal can be thought of as the maximal element of the partially ordered setOriginally Posted by roshanhero
. It's really just an ideal
such that if
is an ideal with
then
. It's the 'biggest' ideal.
A quotient ring is just like all the other quotient structures in algebra. Given a two-sided idealwe 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.
Note that the existence of maximal ideals in an infinite ring relies on the Axiom of Choice (see Zorn's Lemma).A maximal ideal can be thought of as the maximal element of the partially ordered set
That is to say, assuming the Axiom of Choice, maximal ideals always exist. If you do not assume it, they may not.
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