Re: Understanding the quotient ring nomenclature

On Jun 20, 4:27 pm, "Tim BandTech.com" <tttppp...@xxxxxxxxx> wrote:
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.

Right, and we construct a new ring called B[X], and it has an
underlying set {red, blue, yellow, pink, green, orange, purple, ...}.
Now, if we define addition and multiplication operations on that new
set such that the addition and multiplication operations meet the ring
axioms, is B[X] a ring? Is B[X] the same ring as B? Is yellow or
purple an element of B?

Thus far nobody has bothered to show me how you
can do anything with X.

If you mean, why would I bother to construct a new ring; what can I do
with it? Well yes, no one has told you that, but that's because you
keep failing to fully grasp that we have indeed created a new ring and
rant about how the whole subject is lies and should be abolished. That
disinclines people from explaining the more involved material that
actually sees such ring put to a wide variety of practical uses.

If you mean "how do I add to multiply X with anything"; we have shown
you, indeed, we've given the other elements of our newly constructed
ring names which make it obvious how to get them by addition and
multiplication operations involving X. It seems to me, however, that
you are not ready to see that yet. You still have an apparent mental
block about X having to _actually_ be some number instead of realising
that sometimes yellow can just be yellow.

The product rules on B[X] are defined back in B, not in B[X].

No. The rules for the multiplication operation in B[X] are defined
when we define B[X]. We do take care to define the rules such that,
for *the* *subset* of elements in B[X] that are elements of B, the
multiplication operation in B[X] conforms to the multiplication
operation in B. Of course in a multiplication involving elements in B
[X] that are not elements of B things can be different to
multiplication in B.

Yes, I still believe that B[X] = B.

See, this is one of those points where a little better phrasing on
your part would result in much less hostility and frustration directed
your way. With less aggravation driving the thread I suspect it would
be much more conducive to learning. So, what should you have said? How
about: "I still don't understand how B[X] is not just B; could you
explain it another way?". Do you see the difference?

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?

Simply an element in it's own right -- why not? A ring is just a set
with operations, and a set is just a collection of whatever is in the
set; elements of sets don't have to be numbers. So when we construct R
[X] we include, in the underlying set, things like c as just a thing
in its own right, quiute distinct from any real number. Then, when we
get "2.1 X - 3.4 X X" we see that it is simply an element of R[X] that
happens to not be some real number.

It came from the ring product rules which
occurred on the ring R of real values.

Perhaps my answer a while ago confused you. When I said that the
operations in a polynomial were ring operations I meant that they were
the addition and multiplication operations of R[X]! I thought I made
that clear since then, but it seems worth repeating. The addition and
multiplication operations that make up a polynomial are the addition
and multiplication operations of the ring R[X]. They are *not* the
addition and multiplication operations of the ring R of real numbers.
There are two different rings here: R and R[X]. Different rings, by
necessity, have different addition and multiplication operations. The
addition and multiplication operations involved in a polynomial are
the addition and multiplication operations of R[X]. Is that clear?

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?

No, because X is not al algebraic placeholder, but simply an element
in its own right. We can have an evaluation map, which is a function
from one ring to another ring which could thought of as "assigning X a
value", but that is just an analogy.

Is X in
B[X]?

Yes, because we define the underlying set of B[X] to be such that it
contains an element X.

Is B in B[X]?

I'm not sure what you mean? Is B a subset of B[X]? Effectively yes,
and with a little abuse of technicalities we can say yes B is even a
subring of B[X]. Technically no, B and B[X] are distinct separate
rings, and there simply exists an injective ring homomorphism from B
to B[X] -- that is, a function that takes each and every element of B
to a distinct element in B[X] in a way that makes the addition and
multiplication operations compatible between the two rings (for the
subset of B[X] that B maps to).

does the magical constraint on X exclude B?

I don't know what magical constraint you speak of, so I really can't
answer this.


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.

Look Tim, you don't have this right, I assure you. You expose
misunderstanding with each post. Don't go thinking you understand it
all perfectly when you don't. That way lies crankhood, and that's
really not good.

I'm going to try something rather different, more to try and provide
some motivation, and I fear it may just confuse you more, so read
through what follows, see if it makes sense and makes you see all of
this in a different light. If it doesn't just ignore it, and probably
walk away and spend some time not thinking about this. I think you've
worn a track in your thinking which has you stuck thinking about all
of this in a way that is wrong, but increasingly hard to break you out
of.

So, a different approach. Let's forget about polynomial rings
completely for a bit and just let all of it slide out of mind. Instead
I'm just going to work solely with nice concrete real numbers, so
everything should be solidly grounded for you. What I want to do is
consider the integers (I'll denote the set of all integers as Z, as
per usual notation). They form a subset of the real numbers, and do so
in a way that addition and multiplication of integers always gives
another integer. Thus, if we restrict ourselves to only addition and
multiplication we can't ever "break out" of the subset of the real
numbers that is the integers. Now, what if we wanted to extend up from
the integers to the real numbers just a little bit at a time. What do
I mean? Well let's say we wanted consider the subset of the real
numbers formed by the integers and *just* *one* real number? As my
real number I've chosen pi, since it is a nice special real number.

So: I want to consider the set Z U {pi}. A problem arises in that as
soon as we try and do addition and multiplication on this set, we find
that if we do any sum or product that involves pi (other than 0 + pi
and 1 * pi) we immediately end up outside the set: for exmaple clearly
2*pi isn't in the set, and neither is 23 + pi. Fine, we'll add those
numbers to the set so we can do addition and multiplication. So now
we're considering the set Z U {pi, 2*pi, 3*pi, ..., 1+pi, 2+pi,
3+pi, ..., 1 + 2*pi, 2 + 2*pi, ..., 1 + 3*pi, 2 +
3*pi, ..., ..., ... }. Okay, well our set just got a lot bigger, but
we are certainly nowhere near the real numbers: everything in the set
is a multiple of pi, or some integer plus a multiple of pi: that's
certainly not all the real numbers. But wait, we missed something:
pi*pi isn't in our enlarged set! Neither is pi^3, nor pi^4 and so on.
With a little work we can find all the extra real numbers so that we
have a set that's closed under addition and multiplication. So what
are those numbers? Well, they'll be things like 1 + 5*pi + 3*pi^3 and
so on; that is, sums of integer multiples of powers of pi. With a
little careful checking you'll find that, despite having added a lot
of extra elements, we still don't have all the real numbers; what we
have is an interesting extension of the integers -- it's what we get
if we take the set of integers union pi, and then add enough elements
to make that closed under addition and multiplication.

Now, by this point you might be getting suspicious given that what we
have is starting to look a bit like polynomials again. Certainly the
elements of our extended natural numbers are all polynomials with
integer coefficients evaluated at pi. That sounds kind of like a
polynomial ring, except we've got pi instead of some mystery X. Now,
we could go through this whole exercise again, extending the integers
by a single element and this time pick e=2.17182818... We would get
the same result: we would have all sums of integer multiples of powers
of e. There are two things to note here: first, while we end up with a
different subset of the reals when we use e instead of pi, as rings
they would be isomorphic -- that is, if you replaced every occurence
of pi with e in one, it would be the other, even respecting the
addition and multiplication; second, this happens because pi and e are
special numbers called transcendental numbers -- if you were to try
this with other real numbers like sqrt(2) you would get different
results (and I encourage you to try it and see; what do you get when
you find the closure under addition and multiplication of Z U sqrt
(2)?).

The first of those points means that extending by any transcendental
number will give you essentially the same ring as a result, so if
we're interested in the behaviour of the ring, and not which
particular subset of the real numbers we happen to pick out, we may as
well write X and not care which real number it is. The second point is
heading toward the fact that this special sort of extension involving
transcendental numbers is, in a way, universal, and you can
reconstruct any other extension from them.

That, however, only deals with extending the integers. What about
extending the real numbers? What about extending a finite field? What
about extending some other ring? Let's start with the real numbers.
Now, when we were extending the integers we had the benefit of the
fact that we knew they were nestled inside the real numbers and that
there were special real numbers that were transcendental that we would
extend by to get those special "univeral" extensions. Starting with
the real numbers we aren't so lucky. They sit inside the complex
numbers, so we could extend by i, but i isn't transcendental, and only
gets us as far as the complex numbers. What we could do, however, is
pretend that the real numbers sit inside some other much bigger number
system, even if we don't know what that number system is. We can fake
knowing by just forming the set R U {po} where we simply assume that
po is a transcendental number in that bigger unknown number system
(and outside of R). Then we just find the closure under addition and
multiplication of R U {po} inside of this mysterious bigger number
system. The result is that we get sums of real number multiples of
powers of po. And we can determine that without having to know what
the bigger number system is, or, indeed, even what po is. This last
point is handy because we _don't_ know what the bigger number system
is, and we certainly don't know what _po_ is.

We can do the same sort of thing with finite fields, and indeed, with
any ring. The catch is that we don't actually necessarily know what
bigger ring the ring we want to extend sits in, nor any transcendental
elements from that bigger ring. The beauty is that we don't have to.
We just have to say that X *is* a transcental element that lives
outside of the ring we want to extend, and we'll end up with a new
bigger ring that has the right universal extension properties that we
desire. The resulting ring will have sums of multiples of powers of X,
and will, indeed, be the polynomial ring constructed from whatever
base ring we were wishing to extend. Also note that the additions and
multiplications in those polynomials are taking place in the larger
ring, not the base ring -- except we don't know the larger ring, so we
simply take the additions and multiplications as taking place in the
polynomial ring, since it is an intermediary between the two. That is,
you will recall that our sums and products of integers multiples of
powers of pi we not integer sums and products, by real number sums and
products. Thus when we extend the real numbers by po and get sums of
real number multiples of powers of po, the sums and products are being
taken in whatever bigger ring contains po, not in the real numbers.

Now, if you take this sort of idea and formalise it up, you end up
with polynomial rings as usually defined. Does this help at all?

.

Quantcast