Talk:Principal ideal domain

ZFC-centrism

It might be worth mentioning that the claim that every PID is a UFD is not generally true in ZF. There's actually a proof that ZF is consistent with the existence of a PID which is not UFD in Hodges' "Model theory". — Preceding unsigned comment added by 79.183.131.103 (talk) 10:35, 17 December 2013 (UTC)

rating update

How about updating the rating of this article? This is no stub. I would say, it is B. What do you think? Spaetzle (talk) 13:59, 16 February 2012 (UTC)

Proof for example

A proof of the example given of a PID that is not an ED would be nice. —Preceding unsigned comment added by Ecorcoran (talk • contribs) 1 December 2004

I'm sure it's given in Wilson, J. C. "A Principal Ring that is Not a Euclidean Ring." Math. Mag. 34-38, 1973... 129.97.45.36 09:42, 14 December 2006 (UTC)

Structure theorem

might be nice to mention the structure theorem for PID's too. —Preceding unsigned comment added by 171.66.56.36 (talk • contribs) 23 March 2007

Which structure theorem? --345Kai (talk) 00:01, 25 March 2009 (UTC)

Definition set off

A defintion of a PID that is set off from the rest of the paragraph would be nice also. —Preceding unsigned comment added by 209.43.8.56 (talk) 15:24, 3 September 2007 (UTC)

leader

I rewrote the leading paragraphs: they contained a lot of stuff about rings and ideals in general: this is not the place for that. I put stuff more pertinent to PIDs, instead.--345Kai 04:43, 19 October 2007 (UTC)

String of class inclusions is Dedekindist

I really don't like the following. It singles out the UFD property of PID as opposed to other properties, like one-dimensionality (Dedekind). --345Kai (talk) 23:54, 24 March 2009 (UTC)

Principal ideal domains fit into the following (not necessarily exhaustive) chain of class inclusions:

• Commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields

OK, so I got rid of the string of class inclusions, and replaced it with prose which is less partial.--345Kai (talk) 00:10, 25 March 2009 (UTC)

Is $\mathbb {Z} _{2}^{\infty }$ a Principal ideal domain?

Moved from Wikipedia:Reference desk

 – Hkhk59333(talk) 08:51, 20 May 2010 (UTC)

Is $\mathbb {Z} _{2}^{\infty }$ a Principal ideal domain? Hkhk59333(talk) 08:51, 20 May 2010 (UTC)

This is a question for WP:RD/Math, not here. When you ask it there, make sure you explain what $\mathbb {Z} _{2}^{\infty }$ is supposed to mean. Algebraist 09:07, 20 May 2010 (UTC)

PIRs aren't always commutative

The lead section says that "a principal ideal ring is a nonzero commutative ring whose ideals are principal," but some sources don't require principal ideal rings (PIRs) to be commutative, see Algebraic Structures by Jaap Top theorem II.4.5 for example, where he says every division ring is a PIR, while non-commutative division rings such as the quaternions also exist. The notes/pdf is a bit unclear on the definition of a PIR though, as principal ideals are technically only defined for commutative rings (but left and right ideals are defined for general rings). Yodo9000 (talk) 18:42, 25 April 2024 (UTC)

Indeed, they should not be assumed to be commutative. Here is another source that doesn't require them to be commutative and which is more clear about the definition ("all its right ideal are right principal and all its left ideals are left principal"). I'm not quite sure how to fix the phrase or whether the information should just be removed (apparently it's taken from MathWorld: diff). Filipjack 19:34, 13 May 2024 (UTC)

Since we have an article on PIRs, the details of the definition of PIRs do not belong to this article. Nevertheless, the definitions are so close that readers may be confused. So, I have moved the difference between PIDs and PIRs into a hatnote. D.Lazard (talk) 11:44, 24 June 2024 (UTC)