When are There Infinitely Many Irreducible Elements in a Principal Ideal Domain?
| dc.creator | Zanello, Fabrizio | |
| dc.date | 2004-11-11 | |
| dc.date.accessioned | 2026-07-07T05:14:13Z | |
| dc.date.available | 2026-07-07T05:14:13Z | |
| dc.description | 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. | |
| dc.description | 5 pages (3 pages in the journal) | |
| dc.identifier | https://arxiv.org/abs/math/0411259 | |
| dc.identifier | http://arxiv.org/abs/math/0411259 | |
| dc.identifier | Amer. Math. Monthly 111 (2004), No. 2, 150-152 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/73194 | |
| dc.subject | Commutative Algebra | |
| dc.subject | Rings and Algebras | |
| dc.subject | 13F10 (Primary); 11R27 (Secondary) | |
| dc.title | When are There Infinitely Many Irreducible Elements in a Principal Ideal Domain? | |
| dc.type | text |