For which additive submonoids $M$ of nonnegative rationals is $F[X;M]$ AP?

TL;DR

This paper characterizes which additive submonoids of nonnegative rationals make the monoid domain $F[X;M]$ have the property that irreducible and prime elements coincide, providing a diagram of implications among submonoids.

Contribution

It offers a complete characterization of submonoids $M$ of $ extbf{Q}_+$ where irreducible and prime elements in $F[X;M]$ are the same, with a detailed diagram of implications.

Findings

Identifies submonoids where irreducibles are primes in $F[X;M]$

Provides a diagram illustrating implications between submonoid types

Precisely locates monoids with the irreducible-prime coincidence property

Abstract

We characterize the submonoids $M$ of the additive monoid $\Q_+$ of nonnegative rational numbers for which the irreducible and the prime elements in the monoid domain $F[X;M]$ coincide. We present a diagram of implications between some types of submonoids of $\Q_+$, with a precise position of the monoids $M$ with this property.