On the Subatomicity of Polynomial Semidomains

Felix Gotti () and Harold Polo
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Felix Gotti: MIT, Department of Mathematics
Harold Polo: University of Florida, Department of Mathematics

A chapter in Algebraic, Number Theoretic, and Topological Aspects of Ring Theory, 2023, pp 197-212 from Springer

Abstract: Abstract A semidomain is an additive submonoid of an integral domain that is closed under multiplication and contains the identity element. Although factorizations and divisibility in atomic domains have been systematically investigated for more than 30 years, the same aspects in the more general context of atomic semidomains have been considered just recently. Here we study subatomicity in the context of semidomains; that is, we study semidomains satisfying divisibility properties weaker than atomicity. We mostly focus on the Furstenberg property, which is due to P. Clark and motivated by the work of H. Furstenberg on the infinitude of primes, and the almost atomic and quasi-atomic properties, introduced by J. G. Boynton and J. Coykendall in the context of divisibility in integral domains. We investigate these three properties in the context of semidomains, paying special attention to whether they ascend from a semidomain to its polynomial and Laurent polynomial extensions.

Keywords: Semidomain; Polynomial semidomain; Integral domain; Furstenberg semidomain; Almost atomic semidomain; Quasi-atomic semidomain; Atomic; Almost atomic; Quasi-atomic; Primary: 16Y60, 11C08; Secondary: 20M13, 13F05 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/978-3-031-28847-0_13

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