Talk:Principal ideal ring
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a principal ideal ring is the same thing as a principal ideal domain. so this should be merged with principal ideal domain
No. A principal ideal ring is a ring in which every ideal is principal. A principal ideal domain, on the other hand, is an integral domain in which every ideal is principal. Not every ring is an integral domain (since not all rings are commutative, for a start!), so the two notions are distinct. Merging the two articles would, therefore, be inappropriate. Sullivan.t.j 23:38, 15 November 2006 (UTC)
Contributing noncommutative definitions and examples
[edit]Hello, I like the article, and am not going to alter any of the commutative things, but noncommutative principal ideal rings have been studied for over 70 years by some big names, so I think it deserves to go in too. A few quick things:
- I will shortly go look up an example of a right not left principal ideal ring to include.
- I know the link to serial ring is currently broken, but I am on the verge of moving that article in soon. Similarly for the Bezout ring link, right now it is pointed at Bezout domains only, but like PIR's, there is a non domain version too! I hope to accomodate that into the Bezout domain article, to minimize changes.
- Dr. Clark, (I think it was you who contributed the special PIR stuff) I noticed the references do not define special PIR the way you do, but after a little thought I convinced myself they were equivalent. I like 'commutative local artinian PIR" much better than "commutative PIR with unique prime ideal which is also nilpotent", and I'm a bit baffled as to why Hungerford and the Zariski-Samuel book didn't seem to bring that equivalence up. Do you have any insight?
Rschwieb (talk) 00:18, 18 March 2011 (UTC)
Need of a more elementary example
[edit]I think the article should mention some more elementary example of a principal ring that is not integral. For example, Z/6Z is a commutative non-integral ring, isn't it also an example of a principal ring? — Preceding unsigned comment added by 190.55.25.247 (talk) 02:35, 27 February 2013 (UTC)