Re: Understanding the quotient ring nomenclature

On Jun 16, 2:41 pm, Brian Chandler <imaginator...@xxxxxxxxxxxxx>
wrote:
Tim BandTech.com wrote:
On Jun 16, 9:41 am, Brian Chandler <imaginator...@xxxxxxxxxxxxx>
wrote:
I really meant to ask: do you read books at all? I mean, does your
"understanding" of rings come from how mathematicians have defined
rings, or is it entirely out of your own head?

Well, once again you demonstrate that you never answer questions.,,



Sorry. Again a tiff on nomenclature. Superposition and addition are
the same thing, as is summation, and even integration. Again to bow to
you I apologize and replace superposition with addition.

As always in mathematics, you can call anything anything you like, as
long as you define your terms. You agree to "replace" superposition by
addition -- do you mean you agree you'll use the name "addition" to
refer to the addition operator of a ring, the same as everyone else?

Yes, the ring operator is so carefully built

What does it mean for an operator to be "carefully built"? And "the
ring operator"? Are you aware that a ring has two operations?

Sorry, I can't understand anything in what you wrote. It sounds a bit
like quite passionate poetry, and reminds me I once owned a book of
poetry in Albanian. The two things I remember about it were that there
were a lot of E-umlauts, and a lot of exclamation marks. I could
(literally) see the passion being expended in these exclamation marks,
but unfortunately I had no clue what the words meant.

FWIW, I'll put ** round the words that have no obvious meaning to me.

... as to be *scrutinizing* the *fundamental*
operations of addition and multiplication, though the words sum and
product are *nearby*. The polynomial representation does need to *match*,
and so insistence on a polynomial with 'placeholders' or of
indeterminate form is not valid. Thus any interpretation that *stresses*
unresolvable X is a misnomer within the ring context. Yes, polynomials
are rings, and in particular polynomials with real coefficients are
always going to be real valued.

What the hell does it mean to say "polynomials are rings"? What does
"yes" mean in this context? It looks like the "yes" of agreement, yet
you have already been told that "polynomials are rings" is simply
meaningless -- are you simply asserting that the meaningless is indeed
true (whatever that means)?

You have a problem with my terminology but have not refuted any of the
content.

Uh, no. I have a problem with your "content", because nothing you have
said carries any discernable meaning to me.

Again I stress that if my interpretation is conflicted then there
ought to be content which is directly errant. Likewise for the subject
of real polynomial rings at hand. At some point one book would need to
be chosen to scrutinize, but it seems most presentations consistently
insist on the abstraction of X within the polynomial product to the
point of not obeying the ring product and sum requirements.

Can you explain what "not obeying the ring product and sum
requirements" means? With an actual example. Show a bit of a textbook

"For all a, b in R, the result of the operation a · b is also in
R."
- http://en.wikipedia.org/wiki/Ring_(mathematics)
We could take just one term of a polynomial with real coefficient:
6.1 x x x .
Here we see the product operation. This is the same product operation
that is discussed within the definition of the ring. Thus 6.1 and x
belong to the same domain. 6.1 is a real value (by choice) and so x
must likewise be real valued. If this is not the case then the
definition of ring has been offended with regard to the detail on
product. For some reason there is an insistence on maintaining the
polynomial in a more abstract form, but this abstract form does
nothing to alter the situation. Ultimately these polynomials and each
of their parts are simply values in R. Upon choosing a polynomial in
real coefficients then via that product and sum those polynomials are
real valued. This is in conflict with interpretations like:
"However, in general, X and its powers, X^k, are treated as formal
symbols, not as elements of the field K."
- http://en.wikipedia.org/wiki/Polynomial_ring#The_polynomial_ring
This qualification of formal symbolism is not valid. We are down at a
primitive level where the very product operation was only just built
and here we see that build already offended. I have no idea why they
consider the field K in that quote. I would think that the field X is
more what they ought to have tried to deny.

I'm going to ignore your upper criticism but if there is something
that you feel is important there please feel free to ask again. I
think the crux is down here and I'd like to stay at the crux.
I understand that communication itself is a very serious problem.
I am sorry to be too loose with my language. Thanks for staying on
though.
I also understand that what I am asking you to consider should be
nearly impossible.
Yet I only need a sliver of your consideration and perhaps then upon
validation that sliver will amplify itself.
When I go all the way back early in this thread the same issue was
there. It's just that now I've refined my criticism down to this
simplistic crux. Do you see that as we talk about definitions which
construct a product that we should not so freely cast into it products
of another form? I cannot see how I am miscommunicating this point
here. I think I've gotten it finally nailed down to a tight argument.
There is a conflict within accepted abstract algebra. It is a woozy
sort of thing that leads into crappy conversations like I've had on
this thread. In a strange way I claim this as support of my argument,
for a cleanly constructed system should not allow any contradictory
statements, and if one is made then its error should be readily
apparent.

- Tim

that contains a bit of stuff that does this disobeying.

Brian Chandler
.

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