Re: Understanding the quotient ring nomenclature

Arturo Magidin <magidin@xxxxxxxxxxxxxx> wrote:
"Tim BandTech.com" <tttppp...@xxxxxxxxx> wrote:

In http://en.wikipedia.org/wiki/Quotient_ring#Examples
"Now consider the ring R[X] of polynomials in the variable X with
real coefficients, and the ideal I = (X^2 + 1) consisting of all
multiples of the polynomial X^2 + 1." Here they have neither
specified what R[X] is nor what X is.

Yes, they have: R[X] is the ring of polynomials in the variable X with
real coefficients. Those words have *precise*, specific meanings. That
means that they have in fact *completely specified* what R[X] and X are.

Part of the problem here is that "variable" is an overloaded term.
We're stuck with it for historical reasons. In this context X can
denote any indeterminate, i.e. any element of some ring containing
R that is transcendental (not algebraic) over R. It is better to
avoid this confusion by defining polynomials by their coefficient
sequences, i.e. they are functions N -> R with finite support that
are added pointwise and multiplied by Cauchy product (convolution).
Then X = (0,1,0,0,0...), and X^n is the sequence having 1 in the
n'th place and 0 elsewhere; r = (r,0,0,0...) for constants r in R.
Now the question "what is X?" has a clear and rigorous answer.

When learning about these ring constructions it is essential to
understand that it is not the particular construction that matters
but rather the essential properties, i.e. one should view the objects
as solutions to a universal mapping problem. This yields tremendous
power, e.g. see some of my prior posts [1] on universal techniques.

--Bill Dubuque

[1] http://google.com/groups/search?q=author%3Adubuque+universal+polynomial
.

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