Re: Euclidean ring

From: gauzz (gauzz_at_gauzz.it)
Date: 03/08/05 

Date: Tue, 08 Mar 2005 18:00:17 GMT

"Arturo Magidin" <magidin@math.berkeley.edu> ha scritto nel messaggio
news:d0knhb$1lfl$1@agate.berkeley.edu...
> In article <7NkXd.1025586$35.38116816@news4.tin.it>,
> gauzz <gauzz@gauzz.it> wrote:
>>
>>"Arturo Magidin" <magidin@math.berkeley.edu> ha scritto nel messaggio
>>news:d0kd8d$1isa$1@agate.berkeley.edu...
>>> In article <PyiXd.1024696$35.38092612@news4.tin.it>,
>>
>>>
>>> Are you ->sure<- this is the ring your really want, and not
>>>
>>> { (a + b*sqrt(-19))/2 : a,b in 2Z or a,b in Z-2Z}
>>>
>>> (i.e., a=b (mod 2))?
>>
>>
>>Yes, for sure. This sholuld have been an example that Principal ring !=>
>>Euclidean ring, but in the textbook there is no prove of it.
>
> Yes, for sure what? Which ring?

{ (a + b*sqrt(-19)) : a,b in 2Z or a,b in Z-2Z}

>
> Look at my first reply: it contains a method whereby you can sometimes
> show that a ring is not Euclidean (by showing it does not have a
> universal left divisor). Try it for either the ring I mentioned, or
> the ring you mentioned.

yes, it is very useful. I did not know about the existence of such result.
However, it seems that even the second ring I mentioned (the "correct" ring)
still has no 1. This sounds strange to me, because for "Principal Ring" one
has to assume that the ring is unitary... I'm a bit confused now.

Relevant Pages

  • Re: Euclidean ring
    ... >Euclidean ring, but in the textbook there is no prove of it. ... --- Calvin ...
    (sci.math)
  • Re: Euclidean ring
    ... >> Euclidean ring, but in the textbook there is no prove of it. ... > In Stewart & Tall's textbook there is a proof of it. ...
    (sci.math)