Re: -- minimal elements for certain classes of subrings of a field

On Jan 19, 3:36 am, Muhammad <mzafrul...@xxxxxxx> wrote:
On Jan 19, 2:00 am, quasi <qu...@xxxxxxxx> wrote:



On Sat, 19 Jan 2008 00:04:01 -0800 (PST), Muhammad

<mzafrul...@xxxxxxx> wrote:
On Jan 18, 8:59 pm, quasi <qu...@xxxxxxxx> wrote:
On Fri, 18 Jan 2008 22:02:03 -0500, quasi <qu...@xxxxxxxx> wrote:
On Sat, 19 Jan 2008 02:12:24 +0000 (UTC), magi...@xxxxxxxxxxxxxxxxx
(Arturo Magidin) wrote:

In article <ra82p3lg817f1t6tqn8l189mcnj28hu...@xxxxxxx>,
quasi  <qu...@xxxxxxxx> wrote:
On Fri, 18 Jan 2008 18:25:12 +0000 (UTC), magi...@xxxxxxxxxxxxxxxxx
(Arturo Magidin) wrote:

In article <atq1p312gak5oh9esk4e4mb1m9t5t6e...@xxxxxxx>,
quasi  <qu...@xxxxxxxx> wrote:
On Fri, 18 Jan 2008 17:12:09 +0000 (UTC), magi...@xxxxxxxxxxxxxxxxx
(Arturo Magidin) wrote:

  [...]

For K = k(x), the posets ic(K) and ufd(K) are equal, and every element
is greater or equal to a minimal element, but there is no minimum
element.

Right; no minimum in general. But if there is a minimum for k, say R,
then R[x] will be a minimum for k(x), if I'm not mistaken. So for
Q(x), for example, Z[x] will be a minimum.

No, I don't think that's right.

Z[x] is minimal, but not a minimum.

For example, consider Z[1/x].

Right you are.

Or for that matter, Z[1/(x-c)], where c is an arbitrary rational.

Or Z[x+a]... etc.

Well, Z[x+a] is the same ring as Z[x],

Whoops, sorry -- they are not always the same.

I was thinking, when I read it, that a was intended to be an integer.
In that case, Z[x+a] and Z[x] would be the same.

In general, if a,b are arbitrary rationals, the rings Z[x+a] and
Z[x+b] are the same iff b-a is an integer.

whereas the rings Z[1/(x-a)] are all distinct (although isomorphic).

One can ask whether such minimal integrally closed rings and UFDs,
when they exist for a given field K, are necessarily isomorphic. Ok,
I'll ask that! Note that for K = Q, the answer is clearly yes. For K =
k(x), the answer is also yes, but that proof, though elementary, is
harder.

It is hard to see where you are going with it. Maybe because I am not
too deeply interested in it.

The idea is, given a field, to find a natural subring in which to
settle before we start factoring things. We want to make sure the ring
is big enough to generate the field, but other than that, we want as
small a ring as possible, subject to maintaining unique factorization,
or at least staying integrally closed. We certainly don't want the
whole field, since that destroys all the would be irreducibles. Looked
at another way, we want as few units as possible.

The two prototypes are Q and Q(x).

In each case, there is a natural subring.

For Q, the natural subring is Z, which is clearly minimum among all
UFDs in Q with quotient field Q.

For Q(x), the natural subring is Z[x], which is minimal, but not
minimum, since there are other minimal UFD subrings with quotient
field Q(x) -- for example, Z[1/x] or, more generally, Z[1/(x-a)] for
an arbitrary a in Q.

Thus, instead of starting with a ring and asking about factorization,
I am starting with a field, and trying to squeeze it down to an
appropriate ring.

But it is always good to consider extreme situations and simple cases
first. For example looking at D = \cap R where R \in ic(K) and asking:
Does D have K for a field of fractions?

In general no -- consider K = Q(x). The intersection would just be Z,
which has fraction field Q, not Q(x).

Same for E = \cap R where R \in uf(K).

Nope -- same counterexample.

Good. In general no. You have two cases for D . (1) K = qf(D), in this
case you will only have to study only the integrally closed rings
between D and K, inclusive. (2) If D is not equal to qf(D), D is a
subring that is integrally closed in K (check). Can you talk about
some other property? So the members of   ic(K) are some kind of D
algebras. Now you would have to look into what a D-subalgebra of K has
to do to be just in ic(K). (That may possibly be your "minimal"
member.) Ask the question: Is the set S = {X: X is a D-subalgebra of K
not in ic(K)} inductive with inclusion as a partial order? I wish I
had the mental presence or interest to know the answer right away. But
if the answer is yes you will have some maximal elements in S. (In
this case you will have to see how you can use them.) If the answer is
no, you can think of something else.
We can look into the case of E, later.







It may be useful to read a paper by Alan Loper and F.
Tartarone. The title is something like: A classification of integrally
closed rings of polynomial containing Z[X]. It was a pre-print when I
saw. Hopefully it has appeared or may be in the process of being
published.

Ok, I'll look at it, but the field Q(x) is not interesting for my
questions -- I already know the answers for that field.

Also, there is a difference in point of view. I'm starting with a
field K and going down to some minimal subring satisfying my
conditions, whereas the paper is starting from a particular subring
and going up.

I'm very much interested in whether the minimal subrings generally
exist, and if not, for which fields they do exist. Thus, my questions
relate to a property (as yet unclear) of the field itself.

For a given field K, when the minimal subrings exist, are they unique?
When they exist but are not unique, are they at least isomorphic?

My guess is that many of these questions can be settled in sci.math by
simple proofs or (more likely) counterexamples, depending on the
question.

quasi- Hide quoted text -

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Producing new results takes interest, (valid reason for interest)
patience, techniques, a lot of hard work and at times willingness to
indulge in wild goose chase.
Muhammad- Hide quoted text -

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Dear Quasi,
(1) My D-subalgebras of K are supposed to contain (a copy of) D.
(2) I do not see a proof of S being inductive. Sorry.
(3) It would be helpful if you could give a clear description of
"minimal".
Muhammad
.

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