On GL-domains and the ascent of the IDF property
Victor Gonzalez, Ishan Panpaliya
TL;DR
The paper investigates GL-domains, defined by the property that the product of two primitive polynomials remains primitive, and establishes a prime-like characterization of GL-domains. It then links GL-domains to maximal common divisors in the MCD context, showing equivalences with pre-Schreier and PSP-domains in that setting, and introduces a new GL-domain class via $R = D + xK[x]$ (with $R = \mathbb{Z} + x\mathbb{R}[x]$ as a concrete example). The central result answers an open question by showing that the $IDF$-property does not ascend from a GL-domain to its polynomial extension, constructing a GL-domain with $IDF$ whose extension $R[x]$ fails $IDF$. This clarifies the limits of IDF ascent within GL-domains and highlights new GL-domain constructions distinct from classical GCD- and PSP-domain behavior.
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.