Talk:Integral domain

message: 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)
replies: text: I changed the importance to mid. Jakob.scholbach 21:24, 13 May 2007 (UTC)
thread_title:  rating? 

message: 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)
replies: text: 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)
replies: text: 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)
thread_title:  Entire ring 

message: 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)
replies: text: The above comment seems to me to be an attempt to start a discussion on the suitability of changing practice in use of mathematical terminology, 'not a suggestion about editing this Wikipedia article, which is what this talk page is for. Or have I misunderstood? JamesBWatson (talk) 15:39, 5 March 2009 (UTC)
replies: text: In Dutch and German language, an <I>integral domain</I> is called an <I>integrity domain</I>. Furthermore, the word <I>whole</I> is used instead of <I>integral</I> as "monically algebraic". So the word <I>integral</I> is only used for "the opposite of differentiation". The word <I>integral</I> in <I>integral domain</I> refers somehow to the fraction field construction, I think. — Preceding unsigned comment added by 178.85.217.36 (talk) 15:58, 5 March 2021 (UTC)
thread_title:  uncomfortable with terminology 

message: There is a page domain (ring theory). On the other hand "Integral domain" is a pleonasm, as for most mathematicians "domain" is a synonym of "integral ring". Thus I propose to merge this page into domain (ring theory). If for some rare authors there is a slight difference between these notions, it deserve to be quoted in a single page. D.Lazard (talk) 22:29, 26 February 2012 (UTC)
replies: text: As far as I understand, this article is about commutative rings, while domain (ring theory) is centered on non-commutative examples, with a suitable bibliography. This splitting seems to make sense, does not it ? French Tourist (talk) 18:03, 29 February 2012 (UTC)
replies: text: I'm with the last comment that the split is OK. In a perfect world they would be in the same article, but keeping commutative domains apart seems like a good idea because of the depth of analysis on them that is not paralleled in the noncommutative case. I think a lot of mathematicians will interpret "domain" as "commutative domain", but I've never seen "integral domain" used for a noncommutative domain, so the commutative case is well placed here and "domain" takes care of the noncommutative case. Rschwieb (talk) 19:11, 9 April 2012 (UTC)
replies: text: No, leave Integral Domain as a separate article. I just arrived there from Discriminant, following a link within the sentence, "As the discriminant is a polynomial function of the coefficients, it is defined as soon as the coefficients belong to an integral domain R and, in this case, the discriminant is in R." The aptness of that link would really suffer if Integral Domain were merged with anything else. Dratman (talk) 10:04, 15 July 2012 (UTC)
replies: text: This is not a comment on the merge, but, if the merge were done, integral domain would redirect to domain (ring theory), or a section of it, so the link wouldn't suffer.  I think commutative vs. non-commutative might be an adequate reason to keep them separate, if that is used in the "real world" (i.e., reliable sources), but I'm not sure.  — Arthur Rubin  (talk) 23:40, 10 August 2012 (UTC)
replies: text: No. To merge integral domain to domain is not just wrong but really stupid. These two are can be very different topics. I am not much aware about domain. But integral domain is a frequently used terminology in abstract algebra. Aravind V R (talk) 08:57, 12 September 2012 (UTC)
replies: text: It's been seven months, and I think the consensus has been pretty clearly against a merge. I'm going to take down the template and hope I'm not breaching some time limit I don't know about. Rschwieb (talk) 12:53, 18 September 2012 (UTC)
thread_title:  Domain and integral domain: Merge proposal 

message: An IP user has removed four time the first of the equivalent the equivalent definitions of an integral domain
: , 
with the fallacious argument that it is exactly the same as the second definition:
: .
In fact there are not the same for several reasons: firstly, the first definition is less technical than the second one, as the latter requires to know what is a zero divisor for being understood. Secondly the first definition involves two elements the ring, while the second one involves only one element. Moreover, if they would be the same, they would be formalized by the same logical formula, which is not the case. The formalization of the first definition is
:(\forall x) (\forall y)\; x\neq 0 \;\operatorname{and}\; y \neq 0 \implies xy\neq 0,
while the second definition is 
:(\not\exists x)\; x\neq 0\; \operatorname{and}\; ((\not\exists y)\; xy=0 \; \operatorname{and}\; y\neq 0).
Clearly these are not the same formula, and showing that they are equivalent requires several logical operations.
Thus the first definition must restored. Because of WP:3RR, this should be done either by the IP himself. if his math expertise is sufficient for understanding these explanations, or by a watcher of this article. D.Lazard (talk) 20:55, 3 November 2017 (UTC)
replies: text: I have restored the disputed definition outside the list, because it is the basic definition, and the others are variants that use more WP:TECHNICAL terminology. D.Lazard (talk) 13:04, 5 November 2017 (UTC)
thread_title:  Equivalent definitions: need of a third party intervention