Valuation Dimension of Ring Zn Using Python

  • Authors

    • Arifin S. Institut Teknologi Bandung, Jalan Ganesha No.10, Bandung, Jawa Barat, Indonesia 40132
    • Garminia H.
    2018-06-13
    https://doi.org/10.14419/ijet.v7i4.16094
  • Uniserial dimension, valuation dimension, ring Zn, Python.

  • Abstract

    We counting valuation dimension of the ring Zn. Recently, Ghorbani and Nazemian introduced the notion of a valuation dimension of a commutative rings that measures of how far a commutative ring deviates from being valuation. They have proven that an Artinian ring has finite valuation dimension. Therefore, the ring Zn has finite valuation dimension. In this article, some methode and a tool to determine the valuation dimension of the ring Zn will be provided using Python.

  • References

    1. [1] Atiyah, M. F. and MacDonald, I. G., 1969, Introduction to Commutative Algebra, Addison-Wesley Publication Co., London

      [2] D. H. Salsabiela, Program Python Menentukan Faktorisasi Prima Sebuah Bilangan Bulat, http://intechgram.blogspot.com/2018/03/program-python-menentukan-faktorisasi.html, 29 April 2018

      [3] Eisenbud, D. and Griffith, P., 1971, Serial Rings, J. Algebra, 17, 389-400.

      [4] Facchini, A. and Salce, L., 1990, Uniserial modules: sums and isomorphisms of subquotients, Comm. Algebra 18, 2, 499textendash{}517.

      [5] Faith, C., 1984, The structure of valuation rings, J. Pure Appl. Algebra, 31, 7-27.

      [6] Faith, C., 1986, The structure of valuation rings II, J. Pure Appl. Algebra, 42, 37-43.

      [7] Gallian, J.A., 2017, Contemporary Abstract Algebra, 9th Edition, USA.

      [8] Ghorbani, A. and Nazemian, Z., 2015, On commutative rings with uniserial dimension, Journal of Algebra and Its Aplications, 14, 1, 1-10.

      [9] Huang, H. Algebra Lecture Notes. Auburn University Press, http://www.auburn.edu/huanghu/math5310/ , 2010.

      [10] Kaplansky, I., 1952, Modules over Dedekind rings and valuation rings, Trance. Amer. Math. Soc., 72, 327-340.

      [11] Lam, T. Y. (1991): A First Course in Noncommutative Rings, Grad. Texts in Math., Springer, New York.

      [12] Manis, M., 1969, Valuation on a commutative ring, Proc. Amer. Math. Soc, 20, 193-198

      [13] Nazemian, Z., Ghorbani, A. and Behboodi, M., 2014, Uniserial dimension of modules, J. Algebra, 399, 894-903.

      [14] Samsul A, Hanni G.Y., and Pudji A., 2016, Dimensi Valuasi dari Daerah Ideal Utama, Prosiding Semnas KPA Univ. Sanata Dharma, 9-16.

      [15] Stoll, R. R., 1961, Set Theory and Logic, Dover Publication Inc., New York.

      [16] Warfield, R. B., 1975, Serial rings and finitely presented modules, J. Algebra, 37, 187-222.

      [17] WOLFRAM Demonstrations Project, Subgroup Lattices of Finite Cyclic Groups, http://demonstrations.wolfram.com/SubgroupLatticesOfFiniteCyclicGroups/, 20 Juli 2018

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  • How to Cite

    S., A., & H., G. (2018). Valuation Dimension of Ring Zn Using Python. International Journal of Engineering & Technology, 7(4), 6351-6356. https://doi.org/10.14419/ijet.v7i4.16094

    Received date: 2018-07-23

    Accepted date: 2019-05-22

    Published date: 2018-06-13