Characterizations of Hemirings by the Properties of Their k-Ideals


In this paper we characterize those hemirings for which each k-ideal is idempotent. We also characterize those hemirings for which each fuzzy k-ideal is idempotent. The space of prime k-ideals (fuzzy k-prime k-ideals) is topologized.

Share and Cite:

M. Shabir and R. Anjum, "Characterizations of Hemirings by the Properties of Their k-Ideals," Applied Mathematics, Vol. 4 No. 5, 2013, pp. 753-768. doi: 10.4236/am.2013.45104.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] H. S. Vandiver, “Note on a Simple Type of Algebra in Which Cancellation Law of Addition Does Not Hold,” Bulletin of the American Mathematical Society, Vol. 40, No. 12, 1934, pp. 914-920. doi:10.1090/S0002-9904-1934-06003-8
[2] A. W. Aho and J. D. Ullman, “Introduction to Automata Theory, Languages and Computation,” Addison Wesley, Reading, 1976.
[3] D. B. Benson, “Bialgebras: Some Foundations for Distributed and Concurrent Computation,” Fundamenta Informatica, Vol. 12, 1989, pp. 427-486.
[4] J. H. Conway, “Regular Algebra and Finite Machines,” Chapman and Hall, London, 1971.
[5] K. Glazek, “A Guide to Literature on Semirings and Their Applications in Mathematics and Information Sciences with Complete Bibliography,” Kluwer Academic Publishers, Berlin, 2002.
[6] J. S. Golan, “Semirings and Their Applications,” Kluwer Academic Publishers, Berlin, 1999. doi:10.1007/978-94-015-9333-5
[7] U. Hebisch and H. J. Weinert, “Semirings: Algebraic Theory and Applications in the Computer Science,” World Scientific, Singapore, 1998. doi:10.1142/3903
[8] W. Kuich and A. Salomma, “Semirings, Automata, Languages,” Springer Verlag, Berlin, 1986. doi:10.1007/978-3-642-69959-7
[9] S. Eilenberg, “Automata, Languages and Machines,” Academic Press, New York, 1974.
[10] E. T. Lee and L. A. Zadeh, “Note on Fuzzy Languages,” Information Sciences, Vol. 1, No. 4, 1969, pp. 421-434. doi:10.1016/0020-0255(69)90025-5
[11] M. Henriksen, “Ideals in Semirings with Commutative Addition,” Notices of the American Mathematical Society, Vol. 6, 1958, p. 321.
[12] K. Iizuka, “On the Jacobson Radial of a Semiring,” Tohoku Mathematical Journal, Vol. 11, 1959, pp. 409-421.
[13] D. R. LaTorre, “On h-Ideals and k-Ideals in Hemirings,” Publicationes Mathematicae (Debrecen), Vol. 12, 1965, pp. 219-226.
[14] L. A. Zadeh, “Fuzzy Sets,” Infection Control, Vol. 8, No. 3, 1965, pp. 338-353. doi:10.1016/S0019-9958(65)90241-X
[15] J. Ahsan, K. Saifullah and M. Farid Khan, “Fuzzy Semirings,” Fuzzy Sets Systems, Vol. 60, No. 3, 1993, pp. 309-320. doi:10.1016/0165-0114(93)90441-J
[16] J. Ahsan, “Semirings Characterized by Their Fuzzy Ideals,” Journal of Fuzzy Mathematics, Vol. 6, 1998, pp. 181-192.
[17] M. Akram and W. A. Dudek, “Intuitionistic Fuzzy Left k-Ideals of Semirings,” Soft Computing, Vol. 12, No. 9, 2008, pp. 881-890. doi:10.1007/s00500-007-0256-x
[18] S. I. Baik and H. S. Kim, “On Fuzzy k-Ideals in Semirings,” Kangweon Kyungki Mathematical Journal, Vol. 8, No.2, 2000, pp. 147-154.
[19] T. K. Dutta and B. K. Biswan, “Fuzzy k-Ideals of Semirings,” Bulletin of Calcutta Mathematical Society, Vol. 87, 1995, pp. 91-96.
[20] S. Ghosh, “Fuzzy k-Ideals of Semirings,” Fuzzy Sets Systems, Vol. 95, No. 1, 1998, pp. 103-108. doi:10.1016/S0165-0114(96)00306-5
[21] C. B. Kim and M. Park, “k-Fuzzy Ideals in Semirings,” Fuzzy Sets Systems, Vol. 81, No. 2, 1996, pp. 281-286. doi:10.1016/0165-0114(95)00161-1
[22] J. Zhan and Z. Tan, “T-fuzzy k-Ideals of Semirings,” Scientiae Mathematicae Japonica, Vol. 58, 2003, pp. 597 601.
[23] W. A. Dudek, M. Shabir and R. Anjum, “Characterizations of Hemirings by Their h-Ideals,” Computers & Mathematics with Applications, Vol. 59, No. 9, 2010, pp. 3167-3179. doi:10.1016/j.camwa.2010.03.003
[24] X. Ma and J. Zhan, “On Fuzzy h-Ideals of Hemirings,” Journal of Systems Science and Complexity, Vol. 20, No. 3, 2007, pp. 470-478. doi:10.1007/s11424-007-9043-0
[25] X. Ma and J. Zhan, “Generalized Fuzzy h-Bi-Ideals and h-Quasi-Ideals of Hemirings,” Information Sciences, Vol. 179, No. 9, 2009, pp. 1249-1268. doi:10.1016/j.ins.2008.12.014
[26] Y. Yin, X. Huang, D. Xu and F. Li, “The Characterization of h-Semisimple Hemirings,” International Journal of Fuzzy Systems, Vol. 11, No. 2, 2009, pp. 116-122.
[27] Y. Yin and H. Li, “The Characterizations of h-Hemi regular Hemirings and h-Intra-Hemiregular Hemirings,” Information Sciences, Vol. 178, No. 17, 2008, pp. 3451-3464. doi:10.1016/j.ins.2008.04.002
[28] J. Zhan, “On Properties of Fuzzy Left h-Ideals in Hemirings with t-Norms,” International Journal of Mathematics and Mathematical Sciences, No. 19, 2005, pp. 3127-3144. doi:10.1155/IJMMS.2005.3127
[29] J. Zhan and W. A. Dudek, “Fuzzy h-Ideals of Hemirings,” Information Sciences, Vol. 177, No. 3, 2007, pp. 876-886. doi:10.1016/j.ins.2006.04.005
[30] M. K. Sen and P. Mukhopadhyay, “von Neumann Regularity in Semirings,” Kyungpook Mathematical Journal, Vol. 35, 1995, pp. 249-258.
[31] G. Birkhoff, “Lattice Theory,” American Mathematical Society, Providence, 1954.

Copyright © 2023 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.