2004/11/11 by Fabrizio Zanello
Mathematics · #math.AC #math.RA #msc:13F10 #msc:11R27
published as Amer. Math. Monthly 111 (2004), No. 2, 150-152 · 5 pages (3 pages in the journal)
arxiv created 2004/11/11 · arxiv updated 2009/12/01
It has been a well-known fact since Euclid's time that there exist infinitely many rational primes. Two natural questions arise: In which other rings, sufficiently similar to the integers, are there infinitely many irreducible elements? Is there a unifying algebraic concept that characterizes such rings? The purpose of this note is to place the fact concerning the infinity of primes into a more general context, one that also includes the interesting case of the factorial domains of algebraic integers in a number field. We show that, if A is a P.I.D., then A contains infinitely many (pairwise nonassociate) irreducible elements if and only if every maximal ideal of A[x] has the same (maximal) height.