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From: William Elliot on 1 Aug 2010 00:05
The expression maximal ideal is ambiguous.
Here are two differing notions of maximal ideal.

By Zorn's lemma, if x /= 0, there's a maximal ideal
I with x not in I. Is I a maximal ideal?

Are the two notions the same for rings?
If different, how are they distinguished?
Are there cases when they are the same?

----
From: W. Dale Hall on 1 Aug 2010 01:34
William Elliot wrote:
> The expression maximal ideal is ambiguous.
> Here are two differing notions of maximal ideal.
>
> By Zorn's lemma, if x /= 0, there's a maximal ideal
> I with x not in I. Is I a maximal ideal?
>
> Are the two notions the same for rings?
> If different, how are they distinguished?
> Are there cases when they are the same?
>
> ----


I'd word the notion you've identified a bit
more carefully, namely, as being "maximal
with respect to not containing x as a member".
The unrestricted "maximal ideal" notion is
"maximal with respect to being a proper ideal
of the ring".

I could imagine that they may not be identical,
but haven't tried to come up with an example.

Dale
From: Arturo Magidin on 1 Aug 2010 01:54
On Jul 31, 11:05 pm, William Elliot <ma...(a)rdrop.remove.com> wrote:
> The expression maximal ideal is ambiguous.

No, it's not.

> Here are two differing notions of maximal ideal.

"Maximal" always refers relative to a particular partial order; when
one talks about "maximal ideals" without specifying a poset, it is
understood and agreed upon that one is talking about the poset of
*all* maximal ideals, just like when one talks about "maximal
subgroups" of groups one is refering to maximal in the poset of all
proper subgroups, etc.

> By Zorn's lemma, if x /= 0, there's a maximal ideal
> I with x not in I.  Is I a maximal ideal?

Not necessarily.

But why restrict to x? You have much larger notions of ideals that
avoid specific subsets.

> Are the two notions the same for rings?

There are many notions; each x gives you a different notion; each
multiplicative subset may even give you yet another one.

> If different, how are they distinguished?

"Maximal ideal" refers to ideals that are maximal, in the lattice of
*ALL* proper ideals. If you want to consider some subset of ideals
(such as the collection of all ideals that do not contain a given x),
then you talk about "ideals maximal among those that ..." or some such
expression.


> Are there cases when they are the same?

Yes: when x is a unit; or more generally, when x is not in the
intersection of all maximal ideals.

--
Arturo Magidin
From: Arturo Magidin on 1 Aug 2010 01:55
On Aug 1, 12:34 am, "W. Dale Hall"
<wdunderscorehallatpacbelldotnet(a)last> wrote:
> William Elliot wrote:
> > The expression maximal ideal is ambiguous.
> > Here are two differing notions of maximal ideal.
>
> > By Zorn's lemma, if x /= 0, there's a maximal ideal
> > I with x not in I. Is I a maximal ideal?
>
> > Are the two notions the same for rings?
> > If different, how are they distinguished?
> > Are there cases when they are the same?
>
> > ----
>
> I'd word the notion you've identified a bit
> more carefully, namely, as being "maximal
> with respect to not containing x as a member".
> The unrestricted "maximal ideal" notion is
> "maximal with respect to being a proper ideal
> of the ring".
>
> I could imagine that they may not be identical,
> but haven't tried to come up with an example.

Consider the ring R= Z/4Z; and let x = 2 + 4Z. The only ideal of R
that does not contain x is (0); and the only maximal ideal of R is
(2).

--
Arturo Magidin
From: William Elliot on 1 Aug 2010 05:29
On Sat, 31 Jul 2010, Arturo Magidin wrote:
> On Jul 31, 11:05�pm, William Elliot

>> By Zorn's lemma, if x /= 0, there's a maximal ideal
>> I with x not in I. Is I a maximal ideal?
>
> Not necessarily.
>
> But why restrict to x? You have much larger notions of ideals that
> avoid specific subsets.
>
Just x is all that's needed for the current problem.

>> Are the two notions the same for rings?
>
> There are many notions; each x gives you a different notion; each
> multiplicative subset may even give you yet another one.
>
>> If different, how are they distinguished?
>
> "Maximal ideal" refers to ideals that are maximal, in the lattice of
> *ALL* proper ideals. If you want to consider some subset of ideals
> (such as the collection of all ideals that do not contain a given x),
> then you talk about "ideals maximal among those that ..." or some such
> expression.
>
How would you express my question at the top of this post?

>> Are there cases when they are the same?
>
> Yes: when x is a unit; or more generally, when x is not in the
> intersection of all maximal ideals.
>
If a not in intersection of all maximal ideals and
I is a maximal ideal without a, then I is a maximal ideal.

Proof. Some maximal J with a not in J; J subset I;
I = J is maximal. QED.

Let (R, disjoint union, intersection, nulset) be the
Boolean ring of finite subsets of an infinite set.

Then R doesn't have any maximal ideals nor is there
any (relative) maximal ideal I with x not in I. Ouch!

In a Boolean ring R without units, if a /= 0
is there a prime ideal I with a not in I?

----
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