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

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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.