Talk:Integral domain

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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)

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[edit] 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.

[edit] rating?

Do you think this article is really "top" importance? E.g. group is considered to be only "high" importance and to me a group is way more an important gadget than an integral domain. I would change it to "mid", if no one objects. Jakob.scholbach 01:02, 13 May 2007 (UTC)

I changed the importance to mid. Jakob.scholbach 21:24, 13 May 2007 (UTC)

[edit] Entire ring

In his classic Algebra text, Serge Lang calls integral domains entire rings. From the discussion on pages 91–92 (3rd ed.) it is clear that this terminology is his own. Does anyone else use this terminology? Is it worth mentioning here (and providing a redirct)? -- Fropuff (talk) 07:11, 20 January 2008 (UTC)

This synonym also appears in the entry Glossary of ring theory. It was added (without citing sources) in 2005, but the editor who added it hasn't edited before or since under the same name. Michael Slone (talk) 18:21, 20 January 2008 (UTC)
I did notice that there. I suspect they took the definition from Lang. I think if we are to include the term in Wikipedia it should really enjoy wider usage then in a single textbook. -- Fropuff (talk) 18:27, 20 January 2008 (UTC)

[edit] uncomfortable with terminology

I am a working algebraic number theorist, and I am not so comfortable with the convention that "domain" means a ring without zero divisors and "integral" connotes commutativity. The adjective "integral" already has a meaning in algebra: it describes a certain kind of element of an overring, or of an extension of rings. I think these two quite different usages will be confusing to many.

As a general rule of thumb, one knows in advance whether one is entertaining possibly non-commutative rings or not. I would call a noncommutative ring without zero divisors a "noncommutative domain", or, if the noncommutativity of the rings is understood, just a "domain."

Admittedly the use of integral in abstract algebra is ripe for being overhauled -- more consistent with the geometric terminology would be just to call a commutative ring "integral" if it has no zero divisors: this would make it true that an affine scheme is integral iff it is Spec of an integral ring. But then one needs to come up with some "geometric" terminology for an integral extension. To the best of my knowledge, there is -- strangely -- no geometric word for this. 72.152.146.78 (talk) 07:49, 18 March 2008 (UTC)Plclark