Home Mathematics On the arithmetic of polynomial semidomains
Article
Licensed
Unlicensed Requires Authentication

On the arithmetic of polynomial semidomains

  • Felix Gotti ORCID logo EMAIL logo and Harold Polo
Published/Copyright: July 26, 2023

Abstract

A subset S of an integral domain R is called a semidomain provided that the pairs ( S , + ) and ( S , ) are semigroups with identities. The study of factorizations in integral domains was initiated by Anderson, Anderson, and Zafrullah in 1990, and this area has been systematically investigated since then. In this paper, we study the divisibility and arithmetic of factorizations in the more general context of semidomains. We are specially concerned with the ascent of the most standard divisibility and factorization properties from a semidomain to its semidomain of (Laurent) polynomials. As in the case of integral domains, here we prove that the properties of satisfying the ascending chain condition on principal ideals, having bounded factorizations, and having finite factorizations ascend in the class of semidomains. We also consider the ascent of the property of being atomic and that of having unique factorization (none of them ascends in general). Throughout the paper, we provide several examples aiming to shed some light upon the arithmetic of factorizations of semidomains.


Communicated by Manfred Droste


Award Identifier / Grant number: DMS-1903069

Award Identifier / Grant number: DMS-2213323

Funding statement: During the preparation of this paper, the first author was supported by the NSF awards DMS-1903069 and DMS-2213323, while the second author was supported by the University of Florida Mathematics Department Fellowship.

Acknowledgements

The authors kindly thank an anonymous referee for helpful suggestions.

References

[1] S. Albizu-Campos, J. Bringas and H. Polo, On the atomic structure of exponential Puiseux monoids and semirings, Comm. Algebra 49 (2021), no. 2, 850–863. 10.1080/00927872.2020.1820514Search in Google Scholar

[2] D. D. Anderson, D. F. Anderson and M. Zafrullah, Factorization in integral domains, J. Pure Appl. Algebra 69 (1990), no. 1, 1–19. 10.1016/0022-4049(90)90074-RSearch in Google Scholar

[3] D. D. Anderson, D. F. Anderson and M. Zafrullah, Rings between D [ X ] and K [ X ] , Houston J. Math. 17 (1991), no. 1, 109–129. Search in Google Scholar

[4] D. D. Anderson, D. F. Anderson and M. Zafrullah, Factorization in integral domains. II, J. Algebra 152 (1992), no. 1, 78–93. 10.1016/0021-8693(92)90089-5Search in Google Scholar

[5] D. F. Anderson and F. Gotti, Bounded and finite factorization domains, Rings, Monoids and Module Theory, Springer Proc. Math. Stat. 382, Springer, Singapore (2021), 7–57. 10.1007/978-981-16-8422-7_2Search in Google Scholar

[6] N. R. Baeth, S. T. Chapman and F. Gotti, Bi-atomic classes of positive semirings, Semigroup Forum 103 (2021), no. 1, 1–23. 10.1007/s00233-021-10189-8Search in Google Scholar

[7] N. R. Baeth and F. Gotti, Factorizations in upper triangular matrices over information semialgebras, J. Algebra 562 (2020), 466–496. 10.1016/j.jalgebra.2020.06.031Search in Google Scholar

[8] J. G. Boynton and J. Coykendall, An example of an atomic pullback without the ACCP, J. Pure Appl. Algebra 223 (2019), no. 2, 619–625. 10.1016/j.jpaa.2018.04.010Search in Google Scholar

[9] H. Brunotte, On some classes of polynomials with nonnegative coefficients and a given factor, Period. Math. Hungar. 67 (2013), no. 1, 15–32. 10.1007/s10998-013-2367-8Search in Google Scholar

[10] A. Bu, J. Vulakh and A. Zhao, Length-factoriality and pure irreducibility, Comm. Algebra 51 (2023), no. 9, 3745–3755. 10.1080/00927872.2023.2187629Search in Google Scholar

[11] F. Campanini and A. Facchini, Factorizations of polynomials with integral non-negative coefficients, Semigroup Forum 99 (2019), no. 2, 317–332. 10.1007/s00233-018-9979-5Search in Google Scholar

[12] P. Cesarz, S. T. Chapman, S. McAdam and G. J. Schaeffer, Elastic properties of some semirings defined by positive systems, Commutative Algebra and its Applications, Walter de Gruyter, Berlin (2009), 89–101. 10.1515/9783110213188.89Search in Google Scholar

[13] S. T. Chapman, J. Coykendall, F. Gotti and W. W. Smith, Length-factoriality in commutative monoids and integral domains, J. Algebra 578 (2021), 186–212. 10.1016/j.jalgebra.2021.03.010Search in Google Scholar

[14] S. T. Chapman, F. Gotti and M. Gotti, Factorization invariants of Puiseux monoids generated by geometric sequences, Comm. Algebra 48 (2020), no. 1, 380–396. 10.1080/00927872.2019.1646269Search in Google Scholar

[15] P. M. Cohn, Bezout rings and their subrings, Proc. Cambridge Philos. Soc. 64 (1968), 251–264. 10.1017/S0305004100042791Search in Google Scholar

[16] J. Correa-Morris and F. Gotti, On the additive structure of algebraic valuations of polynomial semirings, J. Pure Appl. Algebra 226 (2022), no. 11, Paper No. 107104. 10.1016/j.jpaa.2022.107104Search in Google Scholar

[17] J. Coykendall and F. Gotti, On the atomicity of monoid algebras, J. Algebra 539 (2019), 138–151. 10.1016/j.jalgebra.2019.07.033Search in Google Scholar

[18] J. Coykendall and W. W. Smith, On unique factorization domains, J. Algebra 332 (2011), 62–70. 10.1016/j.jalgebra.2010.10.024Search in Google Scholar

[19] M. Droste, W. Kuich and H. Vogler, Handbook of Weighted Automata, Springer, Berlin, 2009. 10.1007/978-3-642-01492-5Search in Google Scholar

[20] A. Geroldinger and F. Halter-Koch, Non-Unique Factorizations, Algebraic, Combinatorial and Analytic Theory, Pure Appl. Math. (Boca Raton) 278, Chapman & Hall/CRC, Boca Raton, 2006. 10.1201/9781420003208Search in Google Scholar

[21] A. Geroldinger and Q. Zhong, A characterization of length-factorial Krull monoids, New York J. Math. 27 (2021), 1347–1374. Search in Google Scholar

[22] R. Gilmer, Commutative Semigroup Rings, Chic. Lect. Math., University of Chicago, Chicago, 1984. Search in Google Scholar

[23] R. Gilmer and T. Parker, Divisibility properties in semigroup rings, Michigan Math. J. 21 (1974), 65–86. 10.1307/mmj/1029001210Search in Google Scholar

[24] J. S. Golan, Semirings and Their Applications, Kluwer Academic, Dordrecht, 1999. 10.1007/978-94-015-9333-5Search in Google Scholar

[25] F. Gotti, Geometric and combinatorial aspects of submonoids of a finite-rank free commutative monoid, Linear Algebra Appl. 604 (2020), 146–186. 10.1016/j.laa.2020.06.009Search in Google Scholar

[26] F. Gotti and B. Li, Divisibility and a weak ascending chain condition on principal ideals, preprint (2022), https://arxiv.org/abs/2212.06213. Search in Google Scholar

[27] F. Gotti and B. Li, Divisibility in rings of integer-valued polynomials, New York J. Math. 28 (2022), 117–139. Search in Google Scholar

[28] F. Gotti and B. Li, Atomic semigroup rings and the ascending chain condition on principal ideals, Proc. Amer. Math. Soc. 151 (2023), no. 6, 2291–2302. 10.1090/proc/16295Search in Google Scholar

[29] A. Grams, Atomic rings and the ascending chain condition for principal ideals, Proc. Cambridge Philos. Soc. 75 (1974), 321–329. 10.1017/S0305004100048532Search in Google Scholar

[30] F. Halter-Koch, Finiteness theorems for factorizations, Semigroup Forum 44 (1992), no. 1, 112–117. 10.1007/BF02574329Search in Google Scholar

[31] F. Halter-Koch, Ideal Systems. An Introduction to Multiplicative Ideal Theory, Monogr. Textb. Pure Appl. Math. 211, Marcel Dekker, New York, 1998. Search in Google Scholar

[32] W. J. Heinzer and D. C. Lantz, ACCP in polynomial rings: A counterexample, Proc. Amer. Math. Soc. 121 (1994), no. 3, 975–977. 10.1090/S0002-9939-1994-1232140-9Search in Google Scholar

[33] N. Jiang, B. Li and S. Zhu, On the primality and elasticity of algebraic valuations of cyclic free semirings, Internat. J. Algebra Comput. 33 (2023), no. 2, 197–210. 10.1142/S021819672350011XSearch in Google Scholar

[34] N. Lebowitz-Lockard, On domains with properties weaker than atomicity, Comm. Algebra 47 (2019), no. 5, 1862–1868. 10.1080/00927872.2018.1524003Search in Google Scholar

[35] H. Polo, Factorization invariants of the additive structure of exponential Puiseux semirings, J. Algebra Appl. 22 (2023), no. 3, Paper No. 2350077. 10.1142/S0219498823500779Search in Google Scholar

[36] V. Ponomarenko, Arithmetic of semigroup semirings, Ukrainian Math. J. 67 (2015), no. 2, 243–266. 10.1007/s11253-015-1077-1Search in Google Scholar

[37] M. Roitman, Polynomial extensions of atomic domains, J. Pure Appl. Algebra 87 (1993), no. 2, 187–199. 10.1016/0022-4049(93)90122-ASearch in Google Scholar

[38] M. Roitman, On the atomic property for power series rings, J. Pure Appl. Algebra 145 (2000), no. 3, 309–319. 10.1016/S0022-4049(98)00089-9Search in Google Scholar

[39] J.-L. Steffan, Longueurs des décompositions en produits d’éléments irréductibles dans un anneau de Dedekind, J. Algebra 102 (1986), no. 1, 229–236. 10.1016/0021-8693(86)90138-9Search in Google Scholar

[40] R. J. Valenza, Elasticity of factorization in number fields, J. Number Theory 36 (1990), no. 2, 212–218. 10.1016/0022-314X(90)90074-2Search in Google Scholar

[41] A. Zaks, Half factorial domains, Bull. Amer. Math. Soc. 82 (1976), no. 5, 721–723. 10.1090/S0002-9904-1976-14130-4Search in Google Scholar

[42] A. Zaks, Half-factorial-domains, Israel J. Math. 37 (1980), no. 4, 281–302. 10.1007/BF02788927Search in Google Scholar

[43] A. Zaks, Atomic rings without a.c.c. on principal ideals, J. Algebra 74 (1982), no. 1, 223–231. 10.1016/0021-8693(82)90015-1Search in Google Scholar

[44] S. Zhu, Factorizations in evaluation monoids of Laurent semirings, Comm. Algebra 50 (2022), no. 6, 2719–2730. 10.1080/00927872.2021.2018449Search in Google Scholar

Received: 2022-03-21
Revised: 2023-03-30
Published Online: 2023-07-26
Published in Print: 2023-09-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 4.2.2026 from https://www.degruyterbrill.com/document/doi/10.1515/forum-2022-0091/html?lang=en
Scroll to top button