Possible flaw in the polynomial ring A[X] construction

On Jul 7, 8:13 am, "*** T. Winter" <***.Win...@xxxxxx> wrote:
In article <0763ef47-e94a-4066-b347-80e4994b4...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> =?ISO-8859-1?Q?Mariano_Su=E1rez=2DAlvarez?= <mariano.suarezalva...@xxxxxxxxx> writes:
> On Jul 6, 10:05=A0am, "*** T. Winter" <***.Win...@xxxxxx> wrote:
...
> > Interestingly, (X) is an ideal in A[X], and that is the same as A[X] with
> > A removed.
>
> Not really. Just consider the polynomial 1 + X...

Right, I must have locked out my brain when I wrote that.
--
*** t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland;http://www.cwi.nl/~***/


I won't hold it against you ***. Still, as you are repelled by my own
interpretation perhaps you see the inklings in your own context of a
weak construction here. It has drawn you to make some poor statements.
Likewise Leland has been caught in the trap. I think you two are
probably the finest persistent posters on this thread. I don't really
like it when you go snipping away points of substance in usenet
discussions, but I do accept that you are a capable mathematician, and
I have seen little of that snippage here on this thread.

Is it so hard to believe that in the accumulation of the subject that
errors have crept in? Let's face it, each author does struggle over
their presentation of this subject. Can you imagine being drawn down
to the level of having to define X as a "variable" or "symbol"? Those
are literal quotes within the definition, not mine. These authors are
now forced to take the topic at hand as a digest rather than diverge
over to their own specialized form. Reform at this level may not be
possible. This then leaves such a reformist author an outcast.

I am still thinking about how the reals can be claimed a subset of the
complex plane. This argument can be built many ways, with the one
simplest default being what we seem to have been discussing. Yet any
line that I draw through the complex plane can be labelled a real
line. We would then have a function mapping from a real to a complex
value. However under this thinking the product of a real x with a
complex z
x z
should go through the transform
( f(x) )( z )
where f() provides the transform from a real value to a complex value.
This would then bring the ring products into conflict since the
product on this real subset of the complex plane will not generally
match the transformed product in the complex plane.

I can see that given the simple mapping that this alternate could be
dismissed as irrelevant. Yet this is the more general way to map the
reals into the complex plane and in hindsight the assumption of the
first simplistic translation is overly specific. It was strictly
chosen because it happens to align the product and sum operations so
as to not require the transform. Nonetheless the conversion of set R
to a set C is properly taken through a function.

Having mapped the real line onto the complex plane, then removing that
real line we are left with an open crack through the old plane; at
least this is one interpretation. I see several interpretations here
but set theory as it has been constructed declares that its elements
be unique. This type of set theory may be regarded as flawed since for
instance if I were to declare a variable of a set and a constant of
that set they may never assume the same value since there is only one
such value. Rather, if they can assume the same value then the
uniqueness concept of elements in a set is broken and the plurality of
those elements is granted. Hence it should be considered that the
removal of the real line under discussion from the complex plane
leaves the complex plane fully intact due to the plurality of its
elements. For instance if we construct two intersecting lines in the
plane then remove one line then remove the other line we would be
caught in a symmetrical flop where the order of operations will become
critical. Under ring theory these problems are dismissed where
commutative operators are in use as they are here for real and complex
values. Thus I would argue that the set theory in use for the ring
definition must be the plural form.

I am not a set theory expert yet as I've gone back to the basics I see
this opening and am curious how one goes about dismissing it as a
proper mathematician. Or then again perhaps I am using some obscure
set theoretic approach here which is already well established. This is
aside the topic we have been covering. As a topic abstract algebra
might be taken as a more general but strictly built algebra from which
the grade school algebra will fall out. Yet thus far I see no careful
respect for variables versus constants versus concrete elements in its
discussion. It seems more that these ontological distinctions are
thrown away by these mathematicians who believe that they are doing a
service by abstracting them to an ingorant form.

- Tim
.

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