On GL-domains and the ascent of the IDF property

Abstract

Following the terminology introduced by Arnold and Sheldon back in 1975, we say that an integral domain $D$ is a GL-domain if the product of any two primitive polynomials over $D$ is again a primitive polynomial. In this paper, we study the class of GL-domains. First, we propose a characterization of GL-domain in terms of certain elements we call prime-like. Then we identify a new class of GL-domains. An integral domain $D$ is also said to have the IDF property provided that each nonzero element of $D$ is divisible by only finitely many non-associate irreducible divisors. It was proved by Malcolmson and Okoh in 2009 that the IDF property ascends to polynomial extensions when restricted to the class of GCD-domains. This result was recently strengthened by Gotti and Zafrullah to the class of PSP-domains. We conclude this paper by proving that the IDF property does not ascend to polynomial extensions in the class of GL-domains, answering an open question posed by Gotti and Zafrullah.