Re: JSH: Now a change

From: William Hughes (wpihughes_at_hotmail.com)
Date: 10/15/04 

Date: 14 Oct 2004 21:18:02 -0700

jstevh@msn.com (James Harris) wrote in message news:<3c65f87.0410141349.77583757@posting.google.com>...
> brianscsmith@yahoo.com (Brian Smith) wrote in message news:<12f59340.0410140521.709fc57c@posting.google.com>...
> > I have a few questions about your 'object ring'.
>
> It helps if you also include the definition. That's ok, I'll give it.
>
> The object ring includes numbers such that 1 and -1 are the only
> rational units and no member of the ring is a factor of any two
> integers that are coprime in the ring of integers.

Which definition is this. I lost count.

Note that (sqrt(3) + sqrt(2)) is a factor of both 2 and 3
in the algebraic integers. So under your latest definition
the algebraic integers are not objects.

Add a strategic "non unit" and you get back to

An object ring is a sub ring of the complex numbers [1], whose intersection
with Q is Z.

Now both the integers and algebraic integers are object rings.
Too bad there is no largest object ring.

>
> For instance, you can't have x a member if it is a factor of 2 and 3,
> as 2 and 3 are coprime.
>
> The simplest way to consider it, is as the largest set that includes
> integers such that coprimeness holds for all integers.
>
> Like if you add 1/2 into a ring with integers, notice that now you
> have 2(1/2) = 1, which means that 2 is a unit, which contradicts with
> what 2 is in the ring of integers.
>
> >
> > 1.) Does it contain the algebraic integers as a subset?
>
> Yes.

Oops

>
> >
> > 2.) Are any algebraic numbers other than algebraic integers in your ring?
>
> Necessarily, yes.
>

Name one.

                      -"William Hughes"

[1] I see you have decided to include trancendentals after all.

Relevant Pages

  • Re: JSH: Simpler way to the exclusionary rings with asymptotic series
    ... non- convergence and what it means with rings like the ring of algebraic ... BUT in the ring of algebraic integers, ... 'object ring' cannot be closed under infinite sums. ...
    (sci.math)
  • Re: JSH: Now the fun part
    ... and remain in that ring. ... To be a unit in the ring of algebraic integers a number must be the ... root of some monic polynomial with integer coefficients and a last ... But that requirement does not exist in the object ring for units. ...
    (sci.math)
  • Re: JSH: The "Published" paper he dosent what you to know about.
    ... I'm not sure what he wants from his "object ring", ... In "The Theory of Algebraic Integers, ... Note the "theory of ideals" in the last sentence. ...
    (sci.math)
  • Re: JSH "problem"
    ... (Or in fact with any ring? ... I started following he was already fixated on the algebraic integers. ... The one he's going on forever about in these threads is the delusion ... as a factor in the object ring". ...
    (sci.math)
  • Re: JSH: More on problem, algebraic integers
    ... >> only rational units defines a ring. ... the "object" ring from the ring of algebraic integers. ... specified are satisfied by the ring of algebraic integers. ... So how do you know the "object ring" is a different ring? ...
    (sci.math)
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