Talk:Integral domain/Archive 1

Archive 1

Noncommutative integral domains

I want an example about integral domain is not field

Do you mean in the article, or in general? $\mathbb {Z}$ is the obvious example.--Clipdude 07:01, 19 Nov 2004 (UTC)

So how about adding that an integral domain isn't always assumed to be commutative and what happens when it's not? http://www.ams.org/msc/16Uxx.html --mindspank 08:07, 15 Feb 2005 (UTC)

I am interested in such generalizations, but I've never seen an integral domain defined as such without the commutative condition. A section on "noncommutative generalizations" could provide links to the appropriate articles. - Gauge 18:20, 15 March 2006 (UTC)

Over here, a "noncommuative integral domain" is called "domain", and integral domains are always commutative. I find it curious that the AMS classification disagrees in this regard. Anyway, a search through all the algebra books I have at hand agrees with my interpretation. However, a search on mathscinet *did* reveal a few (14) articles referring to "noncommutative integral domains". Well, in my oppinion, this is just yet another case where we mathematicians can't agree on a single common terminology :-). So for now, I added a note that some people talk about noncommutative integral domains, but that we call those simply "domains". The article on the latter certainly should be extended. Of course a lot of things break in the noncommutative case. - BlackFingolfin 00:03, 3 May 2006 (UTC)

Is there some kind of requirement on the order of the integral domain?

For example, is there an integral domain with exactly 6 elements? -unsigned

There is no requirement on the order. The question for which numbers there exist integral domains with that order is I think very hard to answer. I don't know if there exist integral domains with six elements. Oleg Alexandrov (talk) 03:54, 31 October 2006 (UTC)

Er, a finite integral domain is a field, and finite fields are of prime power order - or have I got that wrong? Septentrionalis 22:02, 1 November 2006 (UTC)

No, you had it absolutly right. The order of a finite field (therefore a finite integral domain) must be a prime power. Conversly, there exists (up to isomorphism) exactly one finite filed of order a prime power. — Preceding unsigned comment added by 142.104.167.153 (talk) 23:37, 1 November 2006 (UTC)