Re: Understanding the quotient ring nomenclature
- From: "Tim BandTech.com" <tttpppggg@xxxxxxxxx>
- Date: Sat, 20 Jun 2009 13:27:10 -0700 (PDT)
On Jun 20, 2:30 am, Fishcake <kiwisqu...@xxxxxxxxxxxxxxx> wrote:
Tim,
Please answer the following questions:
1. How many elements are in B, as a set?
2. How many elements are in B[X], as a set?
3. How does your answer to 1) and 2) compare?
Do you still believe that B[X] = B?
Where the set B is { red, blue } there are two unique elements in the
set. That is not to say that I cannot still instantiate five variables
and ten constants in B. This would make 15 symbollic elements in B and
until we construct a math expression with them we don't know what they
are. Furthermore we know that it is possible to construct expressions
which do not make any sense such as in the reals
X X + 1 = 0 .
You are the ones who have adopted an abstract X to do mathematics with
which can only be characterized as a "variable" or a "symbol". I am
fine with all of mathematics being symbols in actuality, not some sort
of pseudo symbol as putting the word in quotes would suggest. If we
take these words out of their quotes and start looking at X as
something mathematical rather than something magical then we can make
some solid statements. Thus far nobody has bothered to show me how you
can do anything with X. There has been denial that X builds an
infinite dimensional representation while others think of it that way.
I have no idea how to play the game other than to mimic what you all
do with it. But upon claiming it to be a member of a ring then we see
that it must behave within the properties of that ring. The product
rules on B[X] are defined back in B, not in B[X]. I will continue to
take the ring definition seriously. As far as I can tell by making a
pseudo variable you've only confused yourselves and in doing so you
inteperet me as confused. If I pick elements x and b(n) out of the
ring b these are merely placeholders for values. One can try to
declare b a constant but until that constant is specified it means
very little. Thus one might solve for a constant b and up until the
expression evaluates b is quite unknown. We already have this category
of unknown. X seems to have trumped that unknown with an unknowable.
That is voodoo mathematics.
Yes, I still believe that B[X] = B. The point of contention lays on
the interpretation of X. Until you show me how to work with X I cannot
procede. People have told me that one cannot assign X a value. Is this
true? How then in the ideal is it possible to set up an expression
such as
X X + 1 = 0 ?
On the real valued R[X] I could write
2.1 X - 3.4 X X = c.
Whether I pulled this c out of B[X] or out of B would seem to form an
issue but I don't see the difference. In either case c is real here to
me. What else could it be? It came from the ring product rules which
occurred on the ring R of real values.
The ring definitions state very clearly that freedoms exist. What they
say in compound form is that any expression of products and sums of
their elements will yield an element of their set. Thus if the ring is
the reals then it will yield the reals, no matter how complicated the
expression becomes. I now view the construction B[X] as optional but
am trying to use it in order to satiate everyone's insistence that it
is somehow special. Polynomials are well behaved by the ordinary
mathematical means so all of these complicated symbollic constructions
are silly. Fishcake is it ever legal to assign X a value in B? Is X in
B[X]? Is B in B[X]? does the magical constraint on X exclude B? I
honestly don't know what you will tell me here. I'm guessing that when
pushed to answer these questions I might get quite some varied
responses in a poll of the participants on this thread.
Anyway there is no motion on this topic with you. I would prefer to
seek some way of moving about so as to cover the context differently.
So I think the focus on X might be helpful, but also trying to move on
to the ideal. I see what you all are doing with the math and the
resistance on X. Clearly that has to be necessary down the road
somehow. It seems pretty obvious that it is in the ideal that the
action will occur. Do the rules which you have placed down for X get
broken in the ideal? I've gotten some progress thanks to Bernier but
I'll be honest, reading the texts that I have read the ideal makes no
sense at all to me. I have little doubt as to why that is so. I do
sincerely keep some open minded space, but I will not deny my own
misunderstanding. If I cannot understand this subject, then so be it.
I don't require that you all change the subject to suit my way of
looking at things. Yet what I study is very nearby to what you study
so I struggle with this. I am ready to push on and accept the
disagreement fully. I don't need to believe the topic to move on. Here
I am one student who gets to have ten teachers. Some of them are
complete jerks but a few them are pretty decent. Thank you for your
time and your patience.
- Tim
.
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