Re: Understanding the quotient ring nomenclature

On Jun 23, 9:03 am, "Tim BandTech.com" <tttppp...@xxxxxxxxx> wrote:
I'm sorry Leland but this seems to me a large digression that is
taking place.
Rather than a resolution it seems to be ballooning outward. This is
the effect of multitype operations though I have not exposed it here.

The issue of a transcenedental number within the real numbers is a
hindsight issue.
It does not serve as a motivation in the definition of ring. If
anything it obscures it.
In terms of the polynomial itself this is merely an issue of
redundancy that has been pointed out. Redundancy is acceptable to me
and I do not feel motivated or enlightened by a system which would
rely upon a transcendental number in order to safeguard from
redundancy. The ground was built from such simple symbollism as
   1 + 1 = 2
and so the redundant behavior will never be gotten rid of. It is
within the transcendental number and the prime number.

I'll have to study the polynomial away from the abstract algebra
context to understand how you have wound up where you are at. I have
no problem with the ring definition. But the polynomial usage that is
somewhow productive to you does not yield for me in its abstract form.
I see only the set that it was defined on and nothing more. Upon
defining a set for x then the polynomial does some great things. One
thing it does not do is widen out that set into some new superspace.
Then I can use the ring definition comfortably again.

Well it seems you aren't really interested in pursuing this anymore,
and obviously have some private interpretation of what I've been
trying to explain that leads to difficulties. Unfortunately you seem
neither willing to try and work out why this interpretation must be
wrong and develop a more accurate interpretation, nor to explain what
exactly this interpretation is so I can help you change to a more
correct understanding of what I am trying to communicate. This makes
it rather difficult.

As to transcendentals: the question is essentially this -- how can we
extend the integers by a single element so as generate the largest
possible (and hence most general/universal) ring if we take the
closure of the extended set under ring operations (in this case, ring
operations in the reals). This is a valid an interesting question,
because you can then extend again by a different element and build up
toward the real numbers a little bit at a time. We can, of course,
generalise this question: how do we extend any ring by a single
element in a way that will generate the most universal ring if we take
the closure of that extension in some logical way. This generalised
version is what drives the definition of polynomial rings.

As to your apparent inability to see B[X] as anything other than B:
Let me try one more approach as to how to look at things that might
make this issue clear. I'll make this one even more cosy for you: I'll
use your polysign numbers. Now, a caveat: there are some
technicalities here (such as your use of positive real numbers only,
where the polynomial ring uses both positive and negative reals) that
make the description that follows not precisely accurate. It is close,
however, and for at least getting the general idea of the construction
it should be close enough that the differences don't matter and can be
ignored (just be wary of of using this explanation for anything more
exacting than a general idea of what we're talking about).

Now, you often write things like "Sum {s=1 to n} sx" to refer to a
polysign number as well as your more usual -a+b*c#d etc. Give me a
little slack and let me work with that presentation where the " +
" (with space on either side) refers to the "+" in that sum, and
"+x" (where I don't have a space but put a symbol next to the plus
sign) refers to your polysign +. That is, I want to write -a+b*c#d as -
a + +b + *c + #d just to spread things out, and also so I can write
things like s_1a + s_2b + s_3c + s_4d + s_5e + s_6f for polysign
numbers in P(n) for larger n (in that case, presuming I read your
system correctly, P(6)).

Now what I want you to imagine is P(infinity). Of course this runs
into some difficulties (as infinity often does) so we'll have to be a
little careful. To define P(infinity) let's require that we'll only
allow numbers that have finitely many different signs in them. Any
finite sum or product of such numbers will, necessarily, have only
finitely many signs as well, and we can skirt issues with infinity
since any particular number will always have a finite (but potentially
very large) number of signs to deal with. Of course we won't have any
wrapping since the standard identity you use for different P(n) will
be Sum {s=1 to infinity} sx = 0, and the left hand side of that is not
a number in P(infinity) since it has infinitely many different signs.
We'll just have to accept that though. We will at least retain the
usual sum and product rules for signs, and our restriction to only
dealing with those numbers that have finitely many different signs
will ensure we don't have any difficulties with infinity when doing
so.

Now, the P(infinity) I've just described is, essentially, R[X] (with
those earlier caveats about minor differences kept in mind). You see
an element of P(infinity) is something like s_1a + s_2b + s_9c + s37d
(remember we can have a different sign for each natural number, but we
only consider numbers that have a finite number of different signs).
We could equally well call that a + b*x^1 + c*x^8 (note the offset
here, where s_1 is equivalent to x^0). The sums and products under the
alternate naming scheme will work out just as in the standard polysign
naming scheme; try it and see.

So, I presume you can see that P(infinity) is not just the real
numbers? I hope you can see that P(infinity) is kind of the limit of
doing you're different P(n). The thing is, we can recover all of the
smaller P(n) from P(infinity) if we know how, so those of us doing
abstract algebra tend to take P(infinity) as the starting point --
especially because we can recover other things that aren't P(n) for
any n from P(infinity) by the same sort of process we would use to
recover different P(n).

Is any of that helping to pump your intuition about what is going on?

.

Relevant Pages

  • Re: Understanding the quotient ring nomenclature
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  • Re: Understanding the quotient ring nomenclature
    ... addition and two distinct product operations at play: one is the sum ... about the reals or the usual sum and product. ... A RING is an ordered triple,, where S is a nonempty ... Also, N denotes the nonnegative integers, but it will ...
    (sci.math)
  • Re: Understanding the quotient ring nomenclature
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    (sci.math)
  • Re: Understanding the quotient ring nomenclature
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  • Re: Understanding the quotient ring nomenclature
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