# Thread: Maximal Ideal of rings

1. ## Maximal Ideal of rings

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.

2. Originally Posted by roshanhero
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 maximal ideal can be thought of as the maximal element of the partially ordered set $\left(\left\{I:I\subseteq R\text{ is an ideal}\right\},\subseteq\right)$. It's really just an ideal $J$ such that if $J'$ is an ideal with $J\subsetneq J'$ then $J'=R$. It's the 'biggest' ideal.

A quotient ring is just like all the other quotient structures in algebra. Given a two-sided ideal $I$ we can partition $R$ into equivalence classes where the equivalence relation is $a\sim b\Leftrightarrow a-b\in I$ and we usually denote $[a]$ by the coset notation $a+I$. We can then define a ring structure on $\left\{r+I:r\in R\right\}$ by $(r+I)+(r'+I)=(r+r')+I$ and $(r+I)(r'+I)=(rr')+ I$.

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.

3. Originally Posted by Drexel28
A maximal ideal can be thought of as the maximal element of the partially ordered set $\left(\left\{I:I\subseteq R\text{ is an ideal}\right\},\subseteq\right)$. It's really just an ideal $J$ such that if $J'$ is an ideal with $J\subsetneq J'$ then $J'=R$. It's the 'biggest' ideal.
Note that the existence of maximal ideals in an infinite ring relies on the Axiom of Choice (see Zorn's Lemma).

That is to say, assuming the Axiom of Choice, maximal ideals always exist. If you do not assume it, they may not.

4. Originally Posted by Swlabr
Note that the existence of maximal ideals in an infinite ring relies on the Axiom of Choice (see Zorn's Lemma).

That is to say, assuming the Axiom of Choice, maximal ideals always exist. If you do not assume it, they may not.
I thought this was clear(ish) by my comment about being maximal elements in the poset.

5. Originally Posted by Drexel28
I thought this was clear(ish) by my comment about being maximal elements in the poset.
Sure, but you still need to mention Zorn's lemma. It is standard to assume maximal ideals exist, but you still need assume it. The words Zorn, choice, well-ordered or assume didn't appear in your post...