The Pure and Applied Mathematics (한국수학교육학회지시리즈B:순수및응용수학)

Korean Society of Mathematical Education (한국수학교육학회)
권호 표지
DOI QR코드

DOI QR Code

STUDY ON CLEAN ORDERED RINGS DERIVED FROM CLEAN ORDERED KRASNER HYPERRINGS

  • Received : 2017.07.18
  • Accepted : 2018.03.08
  • Published : 2018.05.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we introduce the notion of a clean ordered Krasner hyperring and investigate some properties of it. Now, let (R, +, ${\cdot}$, ${\leq}$) be a clean ordered Krasner hyperring. The following is a natural question to ask: Is there a strongly regular relation ${\sigma}$ on R for which $R/{\sigma}$ is a clean ordered ring? Our motivation to write the present paper is reply to the above question.

Keywords

References

  1. R. Ameri & A. Kordi: Clean multiplicative hyperrings. Ital. J. Pure Appl. Math. 35 (2015), 625-636.
  2. T. Amouzegar & Y. Talebi: On clean hyperrings. Journal of Hyperstructures 4 (2015), no. 1, 1-10.
  3. D.D. Anderson & V.P. Camillo: Commutative rings whose elements are a sum of a unit and an idempotent. Comm. Algebra 30 (2002), no. 7, 3327-3336. https://doi.org/10.1081/AGB-120004490
  4. A. Asokkumar: Derivations in hyperrings and prime hyperrings. Iran. J. Math. Sci. Inform. 8 (2013), no. 1, 1-13.
  5. P. Corsini & V. Leoreanu: Applications of hyperstructure theory. Advances in Mathematics, Kluwer Academic Publishers, Dordrecht, 2003.
  6. B. Davvaz: Semihypergroup theory. Elsevier, 2016.
  7. B. Davvaz & V. Leoreanu-Fotea: Hyperring theory and applications. International Academic Press, USA, 2007.
  8. B. Davvaz & S. Omidi: Basic notions and properties of ordered semihyperrings. Categ. General Alg. Structures Appl. 4 (2016), no. 1, 43-62.
  9. B. Davvaz & A. Salasi: A realization of hyperrings. Comm. Algebra 34 (2006), no. 12, 4389-4400. https://doi.org/10.1080/00927870600938316
  10. B. Davvaz, P. Corsini & T. Changphas: Relationship between ordered semihypergroups and ordered semigroups by using pseudoorder. European J. Combinatorics 44 (2015), 208-217. https://doi.org/10.1016/j.ejc.2014.08.006
  11. Z. Gu & X. Tang: Ordered regular equivalence relations on ordered semihypergroups. J. Algebra 450 (2016), 384-397. https://doi.org/10.1016/j.jalgebra.2015.11.026
  12. D. Heidari & B. Davvaz: On ordered hyperstructures. Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 73 (2011), no. 2, 85-96.
  13. M. Krasner: A class of hyperrings and hyperfields. Intern. J. Math. and Math Sci. 6 (1983), no. 2, 307-312. https://doi.org/10.1155/S0161171283000265
  14. F. Marty: Sur une generalization de la notion de groupe. $8^{iem}$ Congres Math. Scandinaves, Stockholm, 1934, 45-49.
  15. W.K. Nicholson: Lifting idempotent and exchange rings. Trans. Amer. Math. Soc. 229 (1977), 269-278. https://doi.org/10.1090/S0002-9947-1977-0439876-2
  16. S. Omidi & B. Davvaz: Foundations of ordered (semi)hyperrings. J. Indones. Math. Soc. 22 (2016), no. 2, 129-147.
  17. S. Omidi & B. Davvaz: A short note on the relation N in ordered semihypergroups. GU J. Sci. 29 (2016), no. 3, 659-662.
  18. S. Omidi, B. Davvaz & P. Corsini: Operations on hyperideals in ordered Krasner hyperrings. An. St. Univ. Ovidius Constanta, Ser. Mat. 24 (2016), no. 3, 275-293.
  19. S. Omidi & B. Davvaz: Contribution to study special kinds of hyperideals in ordered semihyperrings. J. Taibah Univ. Sci. 11 (2017), 1083-1094. https://doi.org/10.1016/j.jtusci.2016.09.001
  20. S. Omidi & B. Davvaz: On ordered Krasner hyperrings. Iranian Journal of Mathematical Sciences and Informatics 12 (2017), no. 2, 35-49.
  21. T. Vougiouklis: Hyperstructures and their representations. Hadronic Press, Florida, 1994.