Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
권호 표지
DOI QR코드

DOI QR Code

On Partitioning and Subtractive Ideals of Ternary Semirings

  • Received : 2010.05.26
  • Accepted : 2010.11.04
  • Published : 2011.03.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we introduce a partitioning ideal of a ternary semiring which is useful to develop the quotient structure of ternary semiring. Indeed we prove : 1) The quotient ternary semiring S/$I_{(Q)}$ is essentially independent of choice of Q. 2) If f : S ${\rightarrow}$ S' is a maximal ternary semiring homomorphism, then S/ker $f_{(Q)}$ ${\cong}$ S'. 3) Every partitioning ideal is subtractive. 4) Let I be a Q-ideal of a ternary semiring S. Then A is a subtractive ideal of S with I ${\subseteq}$ A if and only if A/$I_{(Q{\cap}A)}$ = {q + I : q ${\in}$ Q ${\cap}$ A} is a subtractive idea of S/$I_{(Q)}$.

Keywords

References

  1. Paul J. Allen, A fundamental theorem of homomorphism for semirings, Proc. Amer. Math. Soc., 21(1969), 412-416. https://doi.org/10.1090/S0002-9939-1969-0237575-4
  2. Paul J. Allen, J. Neggers and H. S. Kim, Ideal theory in commutative A-semirings, Kyungpook, Math. Journal, 46(2006), 261-271.
  3. Shahabaddin Ebrahimi Atani, The ideal theory in quotient of commutative semirings, Glasnik Matematicki, 42(62)(2007), 301-308. https://doi.org/10.3336/gm.42.2.05
  4. T. K. Dutta and S. Kar, On Regular Ternary Semirings, Advances in Algebra, Proceedings of the ICM Satellite Conference in Algebra and Related Topics, World Scientific (2003), 343-355.
  5. T. K. Dutta and S. Kar, On The Jacobson Radical of A Ternary Semiring, Southeast Asian Bulletin of Mathematics, 28(1)(2004), 1-13.
  6. T. K. Dutta and S. Kar, On Prime Ideals And Prime Radical of Ternary Semirings, Bull. Cal. Math. Soc., 97(5)(2005), 445-454.
  7. T. K. Dutta and S. Kar, A Note on Regular Ternary Semirings, Kyungpook Math. J., 46(2006), 357-365.
  8. J. S. Golan, Semiring and their Applications, Kluwer Academic publisher Dordrecht, 1999.
  9. Vishnu Gupta and J. N. Chaudhari, On Right ${\pi}$-Regular Semirings, Sarajevo Journal of Mathematics, 2(14)(2006), 3-9.
  10. Vishnu Gupta and J. N. Chaudhari, On Partitioning ideals of Semirings, Kyungpook Math. Journal, 46(2006), 181-184.
  11. S. Kar, Ideal Theory in the Ternary Semiring $\mathbb{Z}^{-}_{0}$, Bull. Malaysian. Math. Sci. Soc., (2), to appear.
  12. D. H. Lehmer, A ternary analogue of abelian groups, American Journal of Mathematics, 59(1932), 329-338.