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

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

2004-11-11

arxiv.org/abs/math/0411259v1

Amer. Math. Monthly 111 (2004), No. 2, 150-152

Mathematics

Commutative Algebra

5 pages (3 pages in the journal)

Scientific paper

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.

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