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

2024/11/01 by Fadinger, Victor, Frisch, Sophie, Nakato, Sarah +2 · 2 citations
#11C08 #11R09 #11S05 #13A05 #13B25 #13F20 #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.2411.01051

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.

Cited by

• On the arithmetic of polynomial ideals
• On the set of atoms and strong atoms in additive monoids of cyclic semidomains

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• A Weaker Notion of Atomicity in Integral Domains
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• The Euclidean criterion for irreducibles
• Spontaneous atomicity for polynomial rings with zero-divisors
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