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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
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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
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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.
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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".
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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
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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.--
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I really don't like the following. It singles out the UFD property of PID as opposed to other properties, like one-dimensionality (Dedekind). --
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I'm sure it's given in Wilson, J. C. "A Principal Ring that is Not a
Euclidean Ring." Math. Mag. 34-38, 1973...
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How about updating the rating of this article? This is no stub. I would say, it is B. What do you think?
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OK, so I got rid of the string of class inclusions, and replaced it with prose which is less partial.--
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A defintion of a PID that is set off from the rest of the paragraph would be nice also.
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Principal ideal domains fit into the following (not necessarily exhaustive) chain of
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A proof of the example given of a PID that is not an ED would be nice.
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might be nice to mention the structure theorem for PID's too.
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Indeed, they should not be assumed to be commutative.
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