PROPERTIES OF HURWITZ POLYNOMIAL AND HURWITZ SERIES RINGS

• Elliott, Jesse (Department of Mathematics California State University at Channel Islands) ;
• Kim, Hwankoo (Division of Computer and Information Engineering Hoseo University)
• Accepted : 2017.09.14
• Published : 2018.05.31

Abstract

In this paper, we study the closedness such as seminomality and t-closedness, and Noetherian-like properties such as piecewise Noetherianness and Noetherian spectrum, of Hurwitz polynomial rings and Hurwitz series rings. To do so, we construct an isomorphism between a Hurwitz polynomial ring (resp., a Hurwitz series ring) and a factor ring of a polynomial ring (resp., a power series ring) in a countably infinite number of indeterminates.

Acknowledgement

Supported by : Hoseo University

References

1. J. A. Beachy and W. D.Weakley, Piecewise Noetherian rings, Comm. Algebra 12 (1984), no. 21-22, 2679-2706. https://doi.org/10.1080/00927878408823127
2. A. Benhissi, Ideal structure of Hurwitz series rings, Beitrage Algebra Geom. 48 (2007), no. 1, 251-256.
3. A. Benhissi, Chain condition on annihilators and strongly Hopfian property in Hurwitz series ring, Algebra Colloq. 21 (2014), no. 4, 635-646. https://doi.org/10.1142/S1005386714000583
4. A. Benhissi and F. Koja, Basic properties of Hurwitz series rings, Ric. Mat. 61 (2012), no. 2, 255-273. https://doi.org/10.1007/s11587-012-0128-2
5. J. W. Brewer, D. L. Costa and K. McCrimmon, Seminormality and root closure in polynomial rings and algebraic curves, J. Algebra 58 (1979), no. 1, 217-226. https://doi.org/10.1016/0021-8693(79)90201-1
6. J. W. Brewer, W. D. Nichols, and, Seminormality in power series rings, J. Algebra 82 (1983), no. 1, 282-284. https://doi.org/10.1016/0021-8693(83)90185-0
7. D. E. Dobbs and M. Roitman, Weak normalization of power series rings, Canad. Math. Bull. 38 (1995), no. 4, 429-433. https://doi.org/10.4153/CMB-1995-062-0
8. G. A. Elliott and P. Ribenboim, Fields of generalized power series, Arch. Math. (Basel) 54 (1990), no. 4, 365-371. https://doi.org/10.1007/BF01189583
9. J. Elliott, Newton basis relations and applications to integer-valued polynomials and q-binomial coefficients, Integers 14 (2014), Paper No. A38, 22 pp.
10. R. Gilmer, Commutative semigroup rings, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1984.
11. R. Gilmer and W. Heinzer, The Laskerian property, power series rings and Noetherian spectra, Proc. Amer. Math. Soc. 79 (1980), no. 1, 13-16. https://doi.org/10.1090/S0002-9939-1980-0560575-6
12. S. Hedayat and E. Rostami, Decomposition of ideals into pseudo-irreducible ideals, Comm. Algebra 45 (2017), no. 4, 1711-1718. https://doi.org/10.1080/00927872.2016.1222410
13. W. Heinzer and J. Ohm, On the Noetherian-like rings of E. G. Evans, Proc. Amer. Math. Soc. 34 (1972), 73-74. https://doi.org/10.1090/S0002-9939-1972-0294316-2
14. W. F. Keigher, On the ring of Hurwitz series, Comm. Algebra 25 (1997), no. 6, 1845-1859. https://doi.org/10.1080/00927879708825957
15. W. F. Keigher and F. L. Pritchard, Hurwitz series as formal functions, J. Pure Appl. Algebra 146 (2000), 291-304. https://doi.org/10.1016/S0022-4049(98)00099-1
16. H. Kim, E. O. Kwon, and T. I. Kwon, Weak normality and strong t-closedness of generalized power series rings, Kyungpook Math. J. 48 (2008), no. 3, 443-455. https://doi.org/10.5666/KMJ.2008.48.3.443
17. J. W. Lim and D. Y. Oh, Composite Hurwitz rings satisfying the ascending chain condition on principal ideals, Kyungpook Math. J. 56 (2016), no. 4, 1115-1123. https://doi.org/10.5666/KMJ.2016.56.4.1115
18. J. W. Lim and D. Y. Oh, Chain conditions on composite Hurwitz series rings, to appear in Open Math..
19. Z. Liu, PF-rings of generalised power series, Bull. Austral. Math. Soc. 57 (1998), no. 3, 427-432. https://doi.org/10.1017/S0004972700031841
20. Z. Liu, Special properties of rings of generalized power series, Comm. Algebra 32 (2004), no. 8, 3215-3226. https://doi.org/10.1081/AGB-120039287
21. C.-P. Lu, Modules and rings satisfying (accr), Proc. Amer. Math. Soc. 117 (1993), no. 1, 5-10. https://doi.org/10.1090/S0002-9939-1993-1104398-7
22. W. K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc. 229 (1977), 269-278. https://doi.org/10.1090/S0002-9947-1977-0439876-2
23. J. Ohm and R. L. Pendleton, Rings with noetherian spectrum, Duke Math. J. 35 (1968), 631-639. https://doi.org/10.1215/S0012-7094-68-03565-5
24. N. Onoda, T. Sugatani, and K. Yoshida, Local quasinormality and closedness type criteria, Houston J. Math. 11 (1985), no. 2, 247-256.
25. G. Picavet, Anodality, Comm. Algebra 26 (1998), no. 2, 345-393. https://doi.org/10.1080/00927879808826134
26. G. Picavet and M. L'Hermitte-Picavet, Morphismes t-clos, Comm. Algebra 21 (1993), no. 1, 179-219. https://doi.org/10.1080/00927879208824555
27. G. Picavet and M. L'Hermitte-Picavet, Anneaux t-clos, Comm. Algebra 23 (1995), no. 7, 2643-2677. https://doi.org/10.1080/00927879508825364
28. M. Picavet-L'Hernitte, Weak normality and t-closedness, Comm. Algebra 28 (2000), no. 5, 2395-2422. https://doi.org/10.1080/00927870008826967
29. R. G. Swan, On seminormality, J. Algebra 67 (1980), no. 1, 210-229. https://doi.org/10.1016/0021-8693(80)90318-X
30. P. T. Toan and B. G. Kang, Krull dimension and unique factorization in Hurwitz polynomial rings, Rocky Mountain J. Math. 47 (2017), no. 4, 1317-1332. https://doi.org/10.1216/RMJ-2017-47-4-1317
31. M. A. Vitulli, Weak normality and seminormality, in Commutative algebra - Noetherian and non-Noetherian perspectives, 441-480, Springer, New York, 2010.
32. H. Yanagihara, Some results on weakly normal ring extensions, J. Math. Soc. Japan 35 (1983), no. 4, 649-661. https://doi.org/10.2969/jmsj/03540649
33. H. Yanagihara, On an intrinsic definition of weakly normal rings, Kobe J. Math. 2 (1985), no. 1, 89-98.