INTEGRAL DOMAINS HAVING NONZERO ELEMENTS WITH INFINITELY MANY PRIME DIVISORS

Abstract. In a factorial domain every nonzero element has only finitely many prime divisors. We study integral domains having nonzero elements with in-finitely many prime divisors. Let D be an integral domain. It is well known that if D is a UFD then every nonzero element has only finitely many prime divisors (see e.g. [G]). This is also true ifD is a Noetherian domain, or more generally, ifD satisfies the ascending chain condition for the principal ideals (ACCP). Indeed, if some nonzero element d ∈ D has infinitely many (mutually non-associate) primes pn, then the principal ideals d/p1 · · · pnD form a strictly ascending chain. Moreover, the element d cannot be written as a product of irreducible elements, say d = a1a2 · · · am, because then each pn is an associate of some aj, a contradiction; so D is not even an atomic domain. We call a domain having nonzero elements with infinitely many prime divisors an IPD domain. Examples of IPD domains are not hard to find. For instance, the ring E of entire functions, that is, complex functions which are analytic in the whole plane, is an IPD domain. Indeed, by [G, page 147, Exercises 16-21], if a, b ar