Re: Understanding the quotient ring nomenclature

On Jun 21, 4:23 pm, "Tim BandTech.com" <tttppp...@xxxxxxxxx> wrote:
On Jun 21, 1:10 am, Leland McInnes <leland.mcin...@xxxxxxxxx> wrote:

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, ...}.

Yikers Leland,
Is this still where
B = { red, blue } ?

Yes, it is. In case you missed it I suggest you go and read through my
post of June 20, 11:24pm (EST) where I explain where all these colours
are coming from. The short answer is that you didn't like it/got
confused when I gave them names that made it clear what elements to
select, so I gave them nice independent names so you can see that I am
actually talking about different _elements_ of a _large_ set.

Assuming your response here is yes then:
What then was the purpose of B?

All the extra elements (other than red and blue) have a relationship
to red and blue (in that they are all the result of some combination
of sums and products of red, blue, and yellow -- that is, we will be
defining & and # in B[X] such that this is the case). Further, the
base ring (in this case B) has a special relationship to the new ring
we create (in this case B[X]); thus if we start with a different base
ring (say the real numbers instead of B) we end up with a very
different new ring. I'm not going to get into the details of all of
that now because it will only get us sidetracked. There's a particular
point I was trying to make here, and there's some hope we can hammer
it home, so I'm going to concentrate on just that one point for the
moment.

Why bother with B at all?

Because it, in some sense, provides the basis (in the general sense,
not the technical mathematical sense) for the new ring we are
constructing. Choose a different ring instead of B and we end up with
a different new ring being constructed.

Simply removing the union of B with B[X] will not eliminate the
elements red and blue since it would appear that they are in B[X]
without the necessity of that union, just as yellow and pink are in B
[X].

I don't understand what this means: it doesn't parse. What are we
"removing" the union B U B[X] from? Why would we want to remove
anything? Why would we form a union of B and B[X] (given that B is a
subset of B[X] by definition)? I'm going to guess there were some
words missing here or something.

Also is my coffee mug in B[X]?
What about a complex number z?
Or is B[X] just a set of colors?

Now we start to get to the point I am hoping to eventually get across
to you. There is a very specific pattern as to what constitutes
elements of B[X]. previously I had been writing out the elements with
names that reflected this pattern, but this seemed to confuse you, as
you didn't seem to realise that they were meant to simply be elements
of the set.

The first thing I need you to be aware of is the fact that in a ring
the _names_ we give to the elements doesn't really matter, what
matters is that we have the right number of elements, and how those
elements _relate_ to one another under the addition and multiplication
of the ring.

So, as an example, let's take B. It was the set {red, blue} with rules
for addition and multiplication as follows:
red + red = red
red + blue = blue
blue + blue = red
red * red = red
red * blue = red
blue * blue = blue
where +, and * are also commutative and associative. Of course red and
blue are just names for the elements of the set. As a ring if I have
(C, + *) with underlying set {0,1} with rules for addition and
multiplication as follows:
0 + 0 = 0
0 + 1 = 1
1 + 1 = 0
0 * 0 = 0
0 * 1 = 0
1 * 1 = 1
where + and * are again commutative and associative then I effectively
have the same ring as B! That is, if I rename red to 0 and blue to 1
in B, I end up with C, and vice versa. Thus it isn't really important
whether I call the element red or 0, what matters is how that element
behaves under addition and multiplication with all the other elements
of the ring. That is, I could equally well have just had {0, blue} and
defined the appropriate + and * operations and I would still have
effectively the same ring. Or I could have the set {pink, watermelon}
and as long as I defined the operations correctly it would still be
effectively the same ring.

The point here is that we give names to elements of the set so we
refer to them, but precisely what those names are doesn't matter as
long as we define the multiplication and addition operations to relate
the elements to one another in the right way.

So, bearing in mind that names are just a useful way of referring to
different elements, the elements of B[yellow] (since X seems to
confuse you) are:

{red, blue, yellow, blue&yellow, yellow#yellow, blue&yellow#yellow,
yellow&yellow#yellow, blue&yellow&yellow#yellow,
yellow#yellow#yellow, ...}

If you're happy with perl style regular expressions, and writing
yellow^n for n-fold multiplication of yellow (i.e. yellow^3 is short
for yellow#yellow#yellow and so on), we want red, blue, yellow, and
things that match: (blue&)?(yellow\^(\d+)&)*yellow\^\d+

So there you have an explicit description of the elements in B
[yellow]. Now, what exactly do I mean when I list expressions like
that? I am saying that when we add blue and yellow we get some element
that is neither red, nor blue, nor yellow. We can call this element
pink if you like. Similarly when I multiply yellow with yellow I get
an element that is not red, nor blue, nor yellow, nor pink. Let's call
that element green. The if I add blue to green I get yet another
element different from any I've described so far which we'll call
orange. Now if I were to add yellow to green I'm going to get yet
another different element again which we'll call purple. Just when you
think doing anything creates entirely new elements though, we have the
following: while adding pink to green yields an entirely new element
that we haven't seen before which we'll call brown, adding yellow to
orange also yields brown, and adding blue to purple yields brown as
well! Thus B[yellow] has a whole lot of new elements in it, but there
are very particular relationships between those new elements under
addition and multiplication in B[yellow].

Thus when I wrote that B[X] had the underlying set {red, blue, yellow,
pink, green, orange, purple, brown, ...} I was trying to underline to
you that we were creating a new ring with a new set of elements, and
there were a lot of different elements in the new ring that weren't in
B.

Now, names aren't important as long as the interrelationships between
elements under addition and multiplication is preserved (that is, we
could call blue 1 instead of blue, and as long as we got the
operations right it didn't matter). Right now specifying all the
interrelationships between the different elements of B[yellow] is hard
because it is never clear what pink#yellow should be, and so on. There
is a fix for this though: we could use names for the elements that
would allow us to work out what pink#yellow should be just by looking
at the names. Thus instead of calling the result of adding blue and
yellow pink, we could instead use the name "blue&yellow" to refer to
that element. Now we can see that pink#yellow is "blue&yellow"#yellow,
and then use the distributive law to get blue#yellow & yellow#yellow,
and finally use the multiplication rule that blue times something is
just the something back again to get yellow & yellow#yellow (which is
an expression that happens to match the regular expression I gave
above). At this point, presuming we were using this "sensible" naming
scheme throughout, we would realise that yellow & yellow#yellow is a
basic independent element of B[yellow] (the one we were previously
calling purple -- the sum of yellow and green).

And this is why I gave the set of element of B[yellow] as

{red, blue, yellow, blue&yellow, yellow#yellow, blue&yellow#yellow,
yellow&yellow#yellow, blue&yellow&yellow#yellow,
yellow#yellow#yellow, ...}

I wanted to use the "sensible" naming scheme that would make it easier
to work out which element a given sum or product was going to work out
to be.

No, note that I am picking out _particular_ expressions involving red,
blue and yellow. Not just any expression will be is a basic element B
[yellow]. Specifically I want only expressions that match the regular
expression

(red|blue|yellow|(blue&)?(yellow\^(\d+)&)*yellow\^\d+)

to be the basic independent elements of B[yellow]. Any other
expression of sums and products of red, blue and yellow *must* work
out to be equal to one (and only one) expression that matches that
regular expression.

Now, if I've explained this sufficiently well you should, hopefully,
be having a minor epiphany and realising that algebraic expressions
involving red, blue and yellow, there is a very specific subset of
expressions that are to distinct independent elements, and all the
rest of the expressions work out to be one of the expressions in that
specific subset. If not, don't worry, I'll try and hit this point
again from a different angle a little lower down in this post.

Actually I am perhaps catching a glimmer of what you are stressing.
Let's see if I can amplify that glimmer.
The specification that are ultimately attempting to consider is
polynomials in coefficients in B on elements in B[X].
As such we could claim the space of such products to be limited by
those coefficient limitations.

Yes, that's the start of the right idea; hopefully what I've written
above has made that more clear and more explicit.

Yet now the puzzle arise for me anyway in an instance:
What is (red)(yellow) ?

Right, well this comes down to the rules we define for multiplication
in B[yellow]. In particular we are going to have to require that red #
yellow = red. Indeed, you'll note that this was one of the rules I
listed in my post of June 20, 11:24pm (EST). Now, a puzzle for you:
can you see why I need to make that rule? It has to do with the fact
that we want red & blue to give the same result as red + blue did in
B, and so on; it may help if you consider renaming the element red and
blue as I did earlier. If you can puzzle this out, it will help you
get a feel for working in rings like B[yellow].

Clearly we have two unique instances in B[X] yet I have no logic for
returning one element in B[X] which is a requirement of the ring
definition. This then returns me to the quesion of what the purpose of
specifying B was in the first place. As I study your questions below
here I see that you are resorting to defining products in B[X]. As
such you have not constructed B[X] yet. You have a partial
construction. The simple perspective is to consider whether you will
redundantly define instances like
red blue
in B[X].

No, part of the point of construct B[yellow] is that the addition and
multiplication operations in B[yellow] should mirror the behaviour of
the addition and multiplication operators in B. Thus since in B we
have red * blue = red, we'll have the rule that red # blue = red in B
[yellow]. Similarly since blue + blue = red in B, we'll have to have
blue & blue = red in B[yellow]. Here's another hint on the puzzle
above: rings have an additive identity element; because we want to be
able to think of B[yellow] as a kind of extension of B, we'll want the
additive identity element of B to also be the additive identity
element of B[yellow].

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?

yes, though its terminology is now obfustated.

Agreed, it is obfuscated. My point was to drive home to you that there
are a lot of new indepdent elements. Hopefully that has come across,
and we can start using the rather less obfuscated naming scheme
described above instead of having to say orange and purple.

Is B[X] the same ring as B?
No.
Is yellow or purple an element of B?
No.

Okay, sorry to stress these points, but I want to be clear that you've
actually understood the point I was trying to make. It seems that you
are at least happy that I'm talking about B[yellow] as a ring that is
quite different, with many more elements, than B. That's a start.

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?

Yes, and I have amplified this question with specifics above here.
This is a multi folded question as to how the elements of B come into B
[X] and whether those elements are in B[X] when B is not unioned into B
[X], thus whether they are redundantly defined.

I'm not sure I understand this question. What do you mean "when B is
not unioned into B[X]"? If perhaps you mean: what will happen if B is
not explicitly a subset of B[X], as I have said is the technically
correct view, then I can provide an answer, albeit a brief one (I
don't want to get away from the important point I'm trying to
emphasise right now): it comes down to ring homomorphisms, which you
may want to look up, but what it amounts to is that, even if B is not
explicitly a subset of B[X], we will require that B[X] have a subset
that is exactly the same size as B, and has exactly the same inter-
relationships under addition and multiplication (within the subset) as
B. That is, there will be some subset of B[X] that you can think of as
B with different names, or a copy of B that works just the same as B.
For now it's probably best to just imagine that B *is* a subset of B
[X], but keep in mind that there are some technical caveats that may
have to be dealt with if things get hairy.

If that wasn't what you intended to ask... I'm not sure what you are
asking, so can't really answer.

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?

Yes, Leland. I'll concede that I should be very nice to you.

It's not a matter of being nice (though it certainly doesn't hurt).
The point is that you are still trying to learn this subject, so when
someone who knows the subject says something that contradicts what you
had been thinking, you should guess that you've misunderstood
something along the way rather than assuming that you know more about
the subject already than the person trying to teach you. That is, if
you spent half as much energy trying to understand how you can
interpret what we say so as to make everything work as you do trying
to interpret what we say so as to make everything not work, you would
learn a whole lot more.

<SNIP>

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?

Geeze, here you've nearly nailed down the issues which I've stressed
above. Yet again I must question whether when we remove the union of R
from R[X] if the values in R are actually still in R[X].

Okay, that last sentence _really_ doesn't parse. The "union of R" with
what? A union needs more than one set, but you have only given one.
Nor am I clear why we want to remove that from R[X]. I think you might
be struggling with the same issue as before -- that you worry about B
being explicitly a subset of B[X] or not. If that's the case then I'll
try and explain a little again, but hopefully we can address this
point in more detail later, after I've got the main point I've been
stressing in this post sorted out.

You can view this two ways. We can take R _as_ _a_ _set_ (ignoring the
addition and multiplication) and have those elements as a subset of R
[X] and define a new set of ring operations on R[X] in such a way
thatthings are compatible with the old operations on R, but also apply
to combinations of the elements we took from R with new elements from R
[X] that weren't in R. Let me make that more explicit: suppose we have
(R, +, *) the real numbers, and we want (R[X], &, #) where I am using
different symbols for addition and multiplication in R[X] so as to
differentiate it from the addition adn multiplication in R. Now, when
we define the addition and multipliaction rules in R[X] we are going
to require the following: if a + b = c in the ring R, then we'll have
a & b = c in R[X], and similarly, if a * b = c in R, then we'll have a
# b = c in R. So, for example, since 3.2 + 2.6 = 5.8 in R, the
operations in R[X] will be defined so that 3.2 & 2.6 = 5.8 in R[X]. Of
course & will be defined over the whole of R[X] and hence different to
+, since 3.2 + X makes no sense in R (since X is not an element of R)
but 3.2 & X will make sense in R[X].

The other way to look at it is that we will require that R[X] have a
special subset such that the subset is closed under addition and
multiplication in R[X] (that is, the sum or product of any two
elements of the subset is always some other element of the special
subset). This special subset will be a *copy* of R -- not the actual
real numbers -- so that the subset behaves just like R when adding or
multiplying within the subset, but the elements of the subset can also
be added or multiplied with all the other elements of R[X] that aren't
in the special subset.

Hopefully you can see how, given that it is the inter-relationships
given by addition and multiplication that matter to a ring, these
different views are in some sense equivalent and interchangeable. If
you can see that, then we might be making progress.

Here to quibble can only lead us back onto that old ground. I grant
you the magic on X. My regurgitant colloidal must be to say yes, Uncle
Leland, though honestly there is a stong acidity as it comes up. The
sentence directly above here does make me want to barf.

No, no, don't just say it is so, try and spend some time thinking
about how you can reconcile in your own head how exactly this will
work. Actually spend some effort trying to figure out how to make this
work instead of just thinking of ways to break it. Once you see how it
works, then you can try and break it -- but coming up with problems
before you properly understand what is being proposed in a large part
of the problem here, because it leads you to misunderstandigns that
are convenient for your efforts to break things, but not helpful for
you to actually learn what it is we are proposing. I'm going to try
and come at the same points from a different direction again below.

<SNIP>

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.

I'm pretty I've gotten all of this. It's Z[pi] I guess isn't it?

Roughly speaking yes. That's the idea I'm heading toward. The point
being that X is not so much magical, as just taken to be a special
kind of element from outside the base ring. If we are dealing with
just the integers then we have the special transcendental numbers
which will do the job of being X and creating a new ring. In general
we aren't so lucky however, and so we just use a handy symbol like X
to denote this special element from whatever larger ring you might
want to imagine there to be.

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 occurrence
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)?).

I have a sense of disbelief that this value sqrt(2) performs
differently than any other constant.

Right, but I think we are getting to an important point here, because
the resulting ring *is* different.

But in terms of redundancy I do see that this value will yield more
redundant terms
1 * (sqrt(2)^2), 2 * sqrt(2)^0
unlike numbers e and pi,

Yes, exactly! With sqtr(2) you find you end up with lots of
expressions that can be simplified back to integers. Thus when we are
adding new elements to the ring we find that sqrt(2)^2 = 2, and so we
don't need to have a new element to deal with sqrt(2)*sqrt(2) since
simplifies down to something already in our set (Z U {sqrt{2}). On the
other hand pi*pi is certainly not an integer. Indeed, with a little
checking you'll find that there are no two polynomials of the form a_0
+ a_1*pi + a_2*pi^2 + ... + a_n pi^n (with the a_i all integers) that
simplify down to the same number... each and every such polynomial is
a different number.

Now, there are expressions that are the same:

2 * (3 + pi)^2 = 2 * (9 + 6 * pi + pi^2) = 18 + 12 * pi + 2 pi^2

but whatever the expression is, we will always be able to at least
simplify it down until it is in the form a_0 + a_1*pi + a_2*pi^2 + ...
+ a_n pi^n. Therefore, since there are no two expressions of the form
a_0 + a_1*pi + a_2*pi^2 + ... + a_n pi^n that evaluate to the same
thing, we need to declare each one of those as a unique distinct
element (which it, because if you evaluate any two such expressions
numerically you'll get different real numbers). On the other hand, we
*only* need to concern ourselves with elements taking exactly that
sort of form, since *any other expression* will simply down to one an
expression of that form (that is, evaluated numerically you'll get a
real number that is equal to the numeric evaluation of some polynomial
of that form). Thus our ring Z[pi] has a unique distinct element for
each expression of the form a_0 + a_1*pi + a_2*pi^2 + ... + a_n pi^n,
and any other expression can be simplified to be one of those
elements.

Now, what of Z[sqrt(2)]? Well obviously sqrt(2)^2 = 2, so we don't
actually have to have sqrt(2)^2 as a distinct element (since it isn't,
it's just 2. Similarly sqrt(2)^3 = 2*sqrt(2), so as long as we've
already got integer multiples of sqrt(2) we've got cubes of sqrt(2)
covered already (since they are just 2*sqrt(2)). And this goes on: we
find that any expression involving integers and sqrt(2), no matter how
complicated, can always be simplified down to something of the form
a_0 + a_1*sqrt(2) (for _0, a_1 integers).

So you see, while the underlying set of Z[pi] has all sorts of
polynomials in it (since each of those polynomials evaluates to a
different real number), Z[sqrt(2)] only has polynomials of the form
a_0 + a_1sqrt(2) (for _0, a_1 integers) in its underlying set, since
any polynomial involving higher powers of sqrt(2) simplifies down to a
polynomial of that form. So Z[pi] has a lot more elements in its
underlying set (since, for instance pi^3 is a distinct element, while
sqrt(2)^3 is really just 2*sqrt(2)).

When we want to form a ring A[X] we want to have X act like pi, and
not like sqrt(2); that is, we want each and every polynomial of the
form a_0 + a_1*X + a_2*X^2 + ... + a_n*X^n to be a distinct unique
element since this gives us the largest possible extension of A that
we can get by taking the closure under addition and multiplication of
A U {X} within some theoretical larger ring containing A. This
"largest possible" idea can actually be formalised as being the
universal one element extension (where "1 element" refers to the fact
that we are initially just adding X and taking the closure, and not
X,Y, and Z and taking the closure).

<SNIP>

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?

Yes Leland. I must grant you X to move on. If you wish I will put up
with a quiz from you in order to progress.
After all, that is the curricular standard which normally satisfies
the subject.

No, no, no. I want you to understand what I am thinking of when I talk
about X and R[X], because it is clear to me that you are not thinking
about the same thing, and that makes any explanations I give of
anything else rather useless because we aren't on the same page. It
isn't worth moving on until you understand.

The quizzes, questions and puzzles aren't to make you regurgitate what
I've said, they are to make you think about what I've said, and
hopefully actually try and work with these things (because practice is
the only way to really get these concepts), and to help me determine
whether you've actually understood what I'm saying. You see you've
said you understand before, but it has become apparent later this we
weren't thinking of the same thing, and hence talking past each other
and amplifying the confusion. I don't want regurgitated answers, I
want you to think about it and tell me what it is you think is the
answer to what I'm asking -- from that I can judge whether we're
talking about the same thing or not, and determine if it is safe to
move on.
.

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