Talk:Prüfer domain
 | This article is rated C-class on Wikipedia's content assessment scale. It is of interest to the following WikiProjects: | ||||||||||
| |||||||||||
PlanetMath
[edit]I've completely rewritten the article and removed the PlanetMath tags as they no longer apply. Arcfrk 18:30, 22 July 2007 (UTC)
Zhat
[edit]The projective limit of the rings Z/nZ (where Z denotes the rational integers and n runs through the natural numbers) is called Prüfer ring by some authors. Is this a Prüfer domain in the sense of the article? 128.176.149.13 (talk) 13:58, 6 February 2009 (UTC)
No: it is not even an integral domain! Using the Chinese Remainder Theorem, one can show that the Prufer ring is the direct product of as p ranges over all prime numbers, so there are plenty of zero divisors. (Perhaps some mention of this should be made in the article?) Plclark (talk) 15:06, 6 February 2009 (UTC)
- Zp is the endomorphism ring of the Prüfer group, and I think but am not sure that the direct product of the Zp is the endomorphism ring of the (rational) circle group Q/Z. I suspect this is where the name comes from.
- I am pretty sure it is a Prüfer ring (not a Prüfer domain). An ideal without zero divisors is contained in one of the Zp, and is isomorphic as R module to Zp. Since Zp is a direct summand of R, it is projective. JackSchmidt (talk) 16:44, 6 February 2009 (UTC)