Re: Understanding the quotient ring nomenclature

On Jun 18, 2:13 pm, "Tim BandTech.com" <tttppp...@xxxxxxxxx> wrote:
On Jun 18, 11:50 am, Brian Chandler <imaginator...@xxxxxxxxxxxxx>
wrote:

Tim BandTech.com wrote:
On Jun 18, 1:14 am, Brian Chandler <imaginator...@xxxxxxxxxxxxx>
wrote:
Tim BandTech.com wrote:

Aside: Is English your first language?
Yes.

Hmm. Then please stop saying things like "definitions are offended",
because they are meaningless. (Not mistranslations as I thought they
might have been.)

I will stand by this language and reinforce the instance below here.
Suppose a to be an element in the ring A and b to be an element in the
ring B. Now I provide two new values
   c = a b
and
   d = a + b

What does "provide" mean here?

In any case: these *may* make sense, if there is some relation between
the ring A and the ring B. For example, if the ring A, with its
operations + and ., is contained in the ring B (with its corresponding
operations) in such a way that the operations of A are the
restrictions of the operations of B, then these two make perfect and
unique sense. In general, however, these are nonsensical expressions,
much like saying "I'm raining you".

These two operators, while they clearly purvey product and sum,

What product? What sum? If you have two distinct rings, then the word
"product" and "sum" have each distinct referents; only if there is
some known relation between the two ring structures would you be able
to refer to them by the single word, but that is not given in your
alleged set-up.

In short, you are talking nonsense. Again.

have
not been provided. Therefore a claim to have provided these operators
via this construction has offended the ring definition.

Rings are not just sets; they are sets equipped with operations. If we
say "A is a ring", then the operations *must* be understood from
context or by prior agreement; otherwise, we are not just being
sloppy, we are in fact not specifying the ring. Much like asking "how
do you say "house' in the language?" and never saying ->which<-
language. It is possible that the "language" is understood: maybe we
are all in a French class, in which case it would be clear that we are
asking about the French word for 'house'. Or maybe we've all
previously agreed to refer to farsi as "the language". But absent that
context or prior agreement, the question becomes meaningless.

Since you provide no context, then you have *not*, in fact, "provided"
the operations. You must *specify* the operations in some way; if you
want, you could say (A,+,x) is a ring and (B,&,#) is another ring. But
of course, if you do that, then your "objection" vanishes, because you
can no pretend longer pretend that you don't get it. Special relations
may be specified between A and B, between + and &, between x and #;
this is what happens when we consider, for example, the integers with
their usual sum and product, and the reals with their usual sum and
product. The reason we can add and multiply integers and reals is
*not* because "the integers are a ring and the reals are a ring". The
reason we can do it is because the set of integers is *contained* in
the set of reals, and the restriction of the real-sum to the subset of
integers *is* the integer-sum, and the restriction of the real-product
to the subset of integers *is* the integer-product. Those are things
that we already know, and that have been previously agreed to: that
the integers are (identified with) a subset of the rationals, that the
rationals are (identified with) a subset of the reals, that the reals
are (identified with) a subset of the complex numbers; and we have
already established (presumably) that the operations on each of the
subsets *agree* with the restriction of the operations on the larger
ones: adding two integers qua integers yields the same answer as if we
add them qua complex numbers. Because all of that is *context* and
*agreed on* prior, then we can simply talk about the integers as being
"in" the complexes, and of a 'single' "addition" and a 'single'
"product".



Definitions can be offended.

As can grammar and language. You are an both an abusive and offensive
individual in that respect, as well as offending and abusing
mathematics and logic.

--
Arturo Magidin
.

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