An application of a generalization of Artin's primitive root conjecture in the theory of monoid rings

Using techniques of algebraic and analytic number theory, we resolve a question on monoid rings posed by Kulosman, et. al., under the assumption of the Generalized Riemann Hypothesis (GRH). Specifically, we show that under an appropriate GRH, for any (rational) prime $p$ the set $E(p)=\{q\text{ prime }\mathrm{\mid }{X}^q-1\text{ factors in }{\mathbb{F}}_p[X;M]\}$, where $M=⟨2,3⟩={\mathbb{N}}_0\setminus \{1\}$, contains a subset with positive natural density. In particular $E(p)\mathrm{\ne }\mathrm{\varnothing }$. This proves that $M$ is not a so-called ``Matsuda monoid’’ of any positive type. For $p=2,3$ this was observed by Kulosman, who provided factorizations of $X^7-1$ and $X^{11}-1$ in ${\mathbb{F}}_2[X;M]$ and ${\mathbb{F}}_3[X;M]$, respectively. Our results explain and reproduce both of these factorizations, as well.