Simple explanation of algebraic integer issue - sci.math

Message from discussion Simple explanation of algebraic integer issue

Bill Dubuque

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More options Oct 4 2003, 6:45 pm

Newsgroups: sci.math

From: Bill Dubuque <w...@nestle.ai.mit.edu>

Date: 04 Oct 2003 18:43:01 -0400

Local: Sat, Oct 4 2003 6:43 pm

Subject: Re: Simple explanation of algebraic integer issue

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Chip Eastham <east...@bellsouth.net> wrote:
>Arturo Magidin <magi...@math.berkeley.edu> wrote:

>> Two ideals A and B are coprime if and only if
>> there does not exist a prime ideal containing both.

>> Two elements r and s of R are "coprime" if and only if
>> the principal ideals they generate, (r) and (s), are coprime.

> Other definitions are possible and probably desirable, in order
> to have greater "separation" between comaximal and coprime.

Indeed, there are various different notions of "coprime" in use.
Thus unless one is a context where the most common alternatives
coincide, e.g. a PID or Bezout domain, one should always define
what "coprime" means. E.g. below are various possible definitions

of a is coprime to b
-----------------

(1) c|a,b => c|1

(2) a,b|c => ab|c

(3) a|bc => a|c

(4) (a) /\ (b) = (ab)

(5) a + b X is a primitive polynomial of degree 1

(6) (a, b) is a primitive ideal

(7) (a, b)^-1 = (1)

(8) (a, b) = (1)

Exercise: what implications hold among these statements
in common classes of domains? Esp. consider these domains:
http://google.com/groups?selm=y8z8yto8k85.fsf%40nestle.ai.mit.edu

-Bill Dubuque

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