 

Felix Gotti and
Bangzheng Li
Divisibility in rings of integervalued polynomials
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Published: 
January 16, 2022. 
Keywords: 
integervalued polynomials, atomic domain, ACCP, ascending chain condition on principal ideals, FFD, finite factorization domain, idfdomain, Furstenberg domain, atomicity, factorization theory. 
Subject: 
Primary: 13A05, 13F15, 13F20; Secondary: 13G05. 


Abstract
In this paper, we address various aspects of divisibility by\break irreducibles in rings consisting of integervalued polynomials. An integral domain is called atomic if every nonzero nonunit factors into irreducibles. Atomic domains that do not satisfy the ascending chain condition on principal ideals (ACCP) have proved to be elusive, and not many of them have been found since the first one was constructed by A. Grams in 1974. Here we exhibit the first class of atomic rings of integervalued polynomials without the ACCP. An integral domain is called a finite factorization domain (FFD) if it is simultaneously atomic and an idfdomain (i.e., every nonzero element is divisible by only finitely many irreducibles up to associates). We prove that a ring is an FFD if and only if its ring of integervalued polynomials is an FFD. In addition, we show that being an idfdomain does not transfer, in general, from an integral domain to its ring of integervalued polynomials. In the same class of rings of integervalued polynomials, we consider further properties that are defined in terms of divisibility by irreducibles, including being CohenKaplansky and being Furstenberg.


Acknowledgements
The first author acknowledges support from the NSF under the award DMS1903069.


Author information
Felix Gotti:
Department of Mathematics
Massachusetts Institute of Technology
Cambridge, MA 02139, USA
fgotti@mit.edu
Bangzheng Li:
Christian Heritage School
Trumbull, CT 06611, USA
libz2003@outlook.com

