Primes and absolutely or non-absolutely irreducible elements in atomic domains

TL;DR

This paper explores the existence of various types of elements such as primes, absolutely irreducible, and irreducible but not absolutely irreducible in atomic integral domains, providing examples for all logical combinations.

Contribution

It constructs examples of atomic integral domains covering all eight possible combinations of these element types, clarifying their coexistence.

Findings

Examples of atomic domains with all combinations of element types

Distinction between primes, absolutely irreducible, and irreducible elements

Insight into the structure of atomic integral domains

Abstract

We give examples of atomic integral domains satisfying each of the eight logically possible combinations of existence or non-existence of the following kinds of elements: 1) primes, 2) absolutely irreducible elements that are not prime, and 3) irreducible elements that are not absolutely irreducible. A non-zero non-unit is called absolutely irreducible (or, a strong atom) if every one of its powers factors uniquely into irreducibles.