Kyungpook Mathematical Journal

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

One-sided Prime Ideals in Semirings

  • Received : 2005.04.20
  • Published : 2007.12.23
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper we define prime right ideals of semirings and prove that if every right ideal of a semiring R is prime then R is weakly regular. We also prove that if the set of right ideals of R is totally ordered then every right ideal of R is prime if and only if R is right weakly regular. Moreover in this paper we also define prime subsemimodule (generalizing the concept of prime right ideals) of an R-semimodule. We prove that if a subsemimodule K of an R-semimodule M is prime then $A_K(M)$ is also a prime ideal of R.

Keywords

References

  1. J. Ahsan, Fully idempotent semirings, Proc. Japan Acad., 69(1993), 185-188.
  2. J. Ahsan, R. Latif ans M. Shabir, Representation of weakly regular semirings by sections in a presheaf, Comm. in Algebra, 21(8)(1993), 2819-2835. https://doi.org/10.1080/00927879308824707
  3. J. Ahsan and Liu Zhongkui, Prime and semiprime acts over monoids with zero, Math. J., Ibaraki University, 33(2001), 9-15. https://doi.org/10.5036/mjiu.33.9
  4. F. Alarcan and D. Polkawska, Fully prime semirings, Kyungpook Math. J., 40(2000), 239-245.
  5. W. D. Blair and H. Tsutsui, Fully prime rings, Communication in Algebra, 22(13)(1994), 5389-5400. https://doi.org/10.1080/00927879408825136
  6. J. Dauns, Prime modules, J.Reine Agnew. Math., 298(1978), 156-181.
  7. S. Feigelstock, Radicals of the semiring of abelian groups, Publ. Math. Debrecen, 27(1980), 89-90.
  8. J. S. Golan, The Theory of Semiring with Applications in Mathematics and Theoretical Computer Science, Pitman Monographs and Surveys in Pure and App. Math., 54, Longman, New York 1992.
  9. F. Hanson, On one sided prime ideals, Pacific J. Math., 58(1)(1975), 79-85. https://doi.org/10.2140/pjm.1975.58.79
  10. K. Koh, On one sided ideals of a prime type, Proc. Amer, Math. Soc., 28(1971), 321-329. https://doi.org/10.1090/S0002-9939-1971-0274488-5