When Are There Infinitely Many Irreducible Elements in a Principal Ideal Domain?

Abstract

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 the present 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 domain (unique factorization domain) of algebraic integers in a number field. We show that, if A is a Principal Ideal Domain (PID, for short), then the fact that A contains infinitely many (pairwise nonassociate) irreducible elements is equivalent to the property that every maximal ideal in the polynomial ring A [x] has the same (maximal) height. We begin by recalling some basic definitions. (We assume that all rings are commutative with an identity element, denoted by 1.) The Jacobson radical J (R) of a ring R is the intersection of all the maximal ideals of R, while the nilradical /O of R is the intersection of all the prime ideals of R. The latter can also be described as the set of all nilpotent elements of R (see [1, Proposition 1.8, p. 5]). The height of a prime ideal P in R is the supremum of the lengths of the chains Po C P1 C ... C Pr = P of prime ideals of R that end at P. The Krull-dimension dim R of R is the supremum of the lengths of all the chains of prime ideals of R, or, equivalently, the supremum of the heights of all the prime ideals P in R. For instance, a field has Krull-dimension 0, while a PID A has Krull-dimension 1, since every prime ideal of A different from (0) has height 1. Finally, two elements a and b of R are associates if there exists a unit u in R such that a = ub. Otherwise, we say that a and b are nonassociates.