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Topic: FLT Discussion: Simplifying
Replies: 65   Last Post: Mar 17, 2001 11:59 PM

 Messages: [ Previous | Next ]
 Michael Hochster Posts: 187 Registered: 12/6/04
Re: FLT Discussion: Simplifying
Posted: Jan 19, 2001 8:51 AM

jstevh@my-deja.com wrote:
: In article <942neb\$dd5\$1@nntp.Stanford.EDU>,
: Michael Hochster <michael@rgmiller.Stanford.EDU> wrote:
:>
:>
:> : (Sort of like if AB = 0, A or B = 0. These people are saying that
: must
:> : be proven, and that it is a "gap" in my proof that I don't do so.)
:>
:> : If so, I'd like them to say that is their position here and we can
: see
:> : if we can't work that one out.
:>
:> Yes, that is my position. I would like an explanation of why
:> it is true that if AB = 0, then A = 0 or B = 0. I grant that
:> this statement is true when A and B are integers. However,
:> I would like you to verify it when A and B are funny things
:> like x + sqrt(-1)y and x - sqrt(-1)y (x, y integers).
:>

: Hey, I've already seen the post where someone says you guys proved that
: AB = 0, when A = 0, or B = 0 by using x^2 + y^2 = 0.

I'm not sure what your point is here. Are you conceding that

: I concede that one could debate the question of whether or not there
: might exist some objects in an infinite ring that could be nonzero and
: multiply times each other to give 0. After all, it's trivally done in
: a finite ring.

Just for future reference, there is standard terminology for this.
A ring in which AB = 0 implies A = 0 or B = 0 is called an
integral domain.

There are simple examples of infinite rings which are not
integral domains. Take ordered pairs of integers with coordinatewise

You wanted to use a "result" (Every infinite ring is an integral
domain) to justify the next step of your argument. But you were
unsure whether the result was true. That means you have a *gap*.
There is no such thing as proof by I-can't-think-of-a-counterexample.
In this case, the result you wanted to use is false.

Mike

Date Subject Author
1/15/01 jstevh@my-deja.com
1/15/01 Dik T. Winter
1/16/01 Charles H. Giffen
1/16/01 jstevh@my-deja.com
1/16/01 Randy Poe
1/18/01 jstevh@my-deja.com
1/18/01 Michael Hochster
1/18/01 Peter Johnston
1/18/01 Randy Poe
1/18/01 Doug Norris
1/16/01 Doug Norris
1/16/01 Randy Poe
1/16/01 Dik T. Winter
1/18/01 jstevh@my-deja.com
1/19/01 Dik T. Winter
1/19/01 Randy Poe
1/20/01 jstevh@my-deja.com
1/20/01 oooF
1/21/01 hale@mailhost.tcs.tulane.edu
1/21/01 Peter Percival
1/21/01 Randy Poe
1/26/01 Franz Fritsche
1/19/01 gus gassmann
1/20/01 jstevh@my-deja.com
1/20/01 Doug Norris
1/26/01 Franz Fritsche
1/16/01 hale@mailhost.tcs.tulane.edu
1/16/01 Randy Poe
1/17/01 hale@mailhost.tcs.tulane.edu
1/18/01 jstevh@my-deja.com
1/19/01 hale@mailhost.tcs.tulane.edu
1/20/01 jstevh@my-deja.com
1/21/01 hale@mailhost.tcs.tulane.edu
1/18/01 Peter Percival
1/19/01 hale@mailhost.tcs.tulane.edu
3/17/01 Ross A. Finlayson
1/16/01 hale@mailhost.tcs.tulane.edu
1/18/01 jstevh@my-deja.com
1/19/01 hale@mailhost.tcs.tulane.edu
1/29/01 jstevh@my-deja.com
1/19/01 Dik T. Winter
1/21/01 Dennis Eriksson
1/15/01 Michael Hochster
1/16/01 jstevh@my-deja.com
1/16/01 Michael Hochster
1/18/01 jstevh@my-deja.com
1/18/01 Peter Percival
1/18/01 Randy Poe
1/19/01 oooF
1/21/01 Dik T. Winter
1/21/01 oooF
1/18/01 Edward Carter
1/19/01 W. Dale Hall
1/19/01 Michael Hochster
1/16/01 Randy Poe
1/16/01 Randy Poe
1/17/01 W. Dale Hall
1/17/01 W. Dale Hall
1/19/01 oooF
1/16/01 Charles H. Giffen
1/16/01 David Bernier
1/16/01 jstevh@my-deja.com
1/18/01 Arthur
1/30/01 plofap@my-deja.com
1/30/01 plofap@my-deja.com
1/30/01 plofap@my-deja.com