Re: Understanding the quotient ring nomenclature

On Jun 16, 5:22 am, Brian Chandler <imaginator...@xxxxxxxxxxxxx>
wrote:
Tim BandTech.com wrote:
Before we get to why, let me pick a nit in your use of language: you
say "the polynomials are not rings" which is fundamentally wrong. The
*set* of *all* polynomials is a ring (note the singular). Saying "the
polynomials are not rings" is saying that the individual polynomials
are not each rings (by the failure to denote the space/set/collection
of polynomials, and by the use of the plural "rings") which is just
nonsense.
Yes, I pushed my criticism too far in my last post. ...

"Criticism" isn't the problem. But you still _haven't_ explained. Do
you understand why saying "the (set of) polynomials form a ring" is
true, and the bits about "polynomials are not rings" is not false, but
just meaningless.

The definition of ring provides superposition and product operators.
These are the same operators in use within the polynomial. Because
superposition requires that each of its elements be in the same domain
then any work in polynomials with real coefficients must obey the
superposition operator. Hence all expressions Sum(a(n)X^n) are real
valued. However the stipulation is often made that X is a placeholder.
This stipulation is in direct conflict with the provision of the
superposition and the product operations, which require that every
element within their use be in the same domain R. This R is
continually confusing since it is meant to mean the ring R here as a
domain, yet if we speak of a polynomial in real coefficients this R is
the real numbers. Hence every polynomial expression is merely a value
in its coefficient. When speaking of a value such as
4.1 x^5 - 3.3 x^6
we do see rich content yet that content is merely a real value. The
representation of polynomials in general via the form
(a0, a1, a2, ...)
is a valid generalized format but content of the result is necessarily
in the same domain as a0 by the definition of ring.

Likewise the usage of unit vectors within this subject will generate a
direct conflict with the definition of ring. This is simply because
the product form
a j
does not contain two elements of the same domain. One is a unit vector
and the other its scalar value. This product does not fit the ring
nomenclature. Yet I do see usages of unit vectors within this subject.
As a high math of the most formal nature (providing the study of the
operators superposition and product) it does not address these
concerns. Ring theory does not accomodate unit vector
representations.

This is mathematics we are discussing, yes? Loose form is what I am
criticized of on this thread yet the tight forms that the subject of
abstract algebra provides are destroyed at its opening. We see good
people like Arturo and Leland picking their words very carefully to
make it through the subject without contradiction, and still I find
contradiction. In a well constructed math this should not be the case.
In an attempt to make a rigorous treatment they enter an infinite
length series as a means of representing the polynomial generally.
This form is not provided for within the ring definition. Because this
series form relies upon superposition and product to meet its
representational requirements and because these operators are provided
for within the definition of a ring their usages beyond the definition
is inappropriate. We cannot use superposition and product so loosely
in a subject which is intended to treat them so carefully as to
actually provide them. Having provided terms in superposition to
contradict superposition by insisting that their domains be isolated
whereas they are actually identical is a direct logical conflict. Such
conflicts are that which mathematics is not. Therefore this is not
serious mathematics.

You speak of sets. Yet what do you have? Will you insist that X is not
a value? What then do you have in the terms of the definition of ring?
The dynamics of the polynomial form are excellent, but those dynamics
are inherently within a specified domain. I do not see that the
distinction of set does anything to change this, for sets are composed
of elements. Generalizations are important but making poor
generalizations and then propagating them so as to hide their
conflicted parts is poor. It leads to conversations like this one.
Many times my direct claims should be directly conflicted if this
subject has integrity. Thus proving my position wrong should be easily
attainable if the subject is well constructed. Whack-a-mole is not an
acceptable mathematical model. It is evidence of a weak construction.
The polynomial interpretation as granting more than the domain of the
coefficient type is incorrect. The sets of results are immense but
that does nothing to change the domain that they exist within.

- Tim


Let me just ask a small question here: was the issue I picked on here
a misunderstanding on your part, or an unfortunate use of language on
your part. You haven't made that clear at all, and it would help if I
actually knew.
A ring is defined as containing elements. Such elements have clear
elemental instances in the reals, the integers, or the complex
numbers, whereas for R[X] there are no such elemental forms.

Well, here you've lost the rails already. Mathematics is *supremely*
about _patterns_. It is not about whether things are "elemental",
"natural" (in the sense of 'not incorporating an internal combustion
engine'), "wooden", or "made of real rubber" (for topology).

Why don't you simply read a book about elementary mathematics --
here's a recommendation:

"A concrete approach to abstract algebra", W W Sawyer. It's good, old,
small, and cheap -- from about $4 here:

http://www.abebooks.com/servlet/SearchResults?an=sawyer&sts=t&tn=conc...

In particular, the discussion of polynomials (over any field) starts
on page 33, and I seem to have previously made a public copy of pp
34-35, which cover the nub of this. Curiously, your problem is
probably that you are one of his "bright" pupils.

http://imaginatorium.org/private/sawyer.gif

HTH (but wonder, somehow)

Brian Chandler

.

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