We all learn in intro abstract algebra that a euclidean domain is a PID. It turns out that the converse is *almost *true. Namely, if one relaxes the definition of a euclidean norm (instead of a euclidean algorithm, you have something a bit weaker) you get something entirely equivalent to being a PID. This is apparently due to Greene in the Monthly, 1997 (and has a quick proof). Now, this material is in ch. 1.

Lang’s *Algebra *(as well as some of his other books, too) has a lot of these kinds of isolated references to scattered results in the literature. Some of these are quite interesting; it is probably worth adding more of these. Doing so will also make the book less “canonical”!

It happens, coincidentally, that we also got a donation on euclidean domains, which has been partially merged in.

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This entry was posted on January 23, 2011 at 1:38 am and is filed under announcements.

February 27, 2011 at 1:39 am

The fact that being a PID is equivalent to the existence of a certain kind of norm which is reminiscent of a Euclidean norm but slightly weaker goes all the way back to Dedekind and was independently rediscovered by Hasse. See Section 8.3 of my survey article on factorization for more information: http://math.uga.edu/~pete/factorization2010.pdf. (I don’t know why Greene’s article was published without any reference to this not-especially-obscure older work. It looks like an oversight to me…)

February 27, 2011 at 3:41 am

Dear Pete, thanks for pointing that out! I’ll fix it when I get some time (which reminds me that I have not worked much on this in the past week or so owing to other commitments — i.e. large loads of homework).